THE
GEOMETRY
OF
FROBENIOIDS
I:
THE
GENERAL
THEORY
Shinichi
Mochizuki
June
2008
We
develop
the
theory
of
Frobenioids,
which
may
be
regarded
as
a
category-theoretic
abstraction
of
the
theory
of
divisors
and
line
bundles
on
models
of
finite
separable
extensions
of
a
given
function
field
or
number
field.
This
sort
of
abstraction
is
analogous
to
the
role
of
Galois
categories
in
Galois
theory
or
monoids
in
the
geometry
of
log
schemes.
This
abstract
category-theoretic
framework
preserves
many
of
the
important
features
of
the
classical
theory
of
divisors
and
line
bundles
on
models
of
finite
separable
extensions
of
a
function
field
or
number
field
such
as
the
global
degree
of
an
arithmetic
line
bundle
over
a
number
field,
but
also
exhibits
interesting
new
phenomena,
such
as
a
“Frobenius
endomorphism”
of
the
Frobenioid
associated
to
a
number
field.
Introduction
§0.
Notations
and
Conventions
§1.
Definitions
and
First
Properties
§2.
Frobenius
Functors
§3.
Category-theoreticity
of
the
Base
and
Frobenius
Degree
§4.
Category-theoreticity
of
the
Divisor
Monoid
§5.
Model
Frobenioids
§6.
Some
Motivating
Examples
Appendix:
Slim
Exponentiation
Index
Chart
of
Types
of
Morphisms
in
a
Frobenioid
Bibliography
Introduction
§I1.
Technical
Summary
§I2.
Abstract
Combinatorialization
of
Arithmetic
Geometry
§I3.
Frobenius
Endomorphisms
of
a
Number
Field
§I4.
Étale-like
vs.
Frobenius-like
Structures
Acknowledgements
2000
Mathematical
Subject
Classification:
Primary
14A99;
Secondary
11G99.
Keywords
and
Phrases:
categories;
Galois
categories;
Frobenius;
monoids;
log
schemes.
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
§I1.
Technical
Summary
In
the
present
paper,
we
introduce
the
notion
of
a
Frobenioid.
The
simplest
kind
of
Frobenioid
“F
M
”
is
the
non-commutative
monoid
given
by
forming
the
“semi-direct
product
monoid”
of
a
given
commutative
monoid
M
with
the
multi-
plicative
monoid
of
positive
integers
N
≥1
[cf.
§0],
where
n
∈
N
≥1
acts
on
M
by
multiplication
by
n;
that
is
to
say,
the
underlying
set
of
F
M
is
the
product
M
×
N
≥1
equipped
with
the
monoid
structure
is
given
as
follows:
if
a
1
,
a
2
∈
M
,
n
1
,
n
2
∈
N
≥1
,
then
(a
1
,
n
1
)
·
(a
2
,
n
2
)
=
(a
1
+
n
1
·
a
2
,
n
1
·
n
2
)
[cf.
Definition
1.1,
(iii)].
For
instance,
when
M
is
taken
to
be
the
additive
monoid
of
nonnegative
integers
Z
≥0
[cf.
§0],
def
we
shall
write
F
=
F
Z
≥0
and
refer
to
F
as
the
standard
Frobenioid.
Note
that
in
general,
any
monoid
[such
as
F
M
,
for
instance]
may
be
thought
of
as
a
category,
i.e.,
the
category
with
precisely
one
object
whose
monoid
of
endomorphisms
is
the
given
monoid.
More
generally,
one
may
start
with
a
“family
of
commutative
monoids”
Φ
on
a
“base
category”
D
[where
Φ,
D
satisfy
certain
properties]
and
form
the
associated
elementary
Frobenioid
F
Φ
by
taking
the
“semi-direct
product”
of
N
≥1
with
Φ
[cf.
Definition
1.1,
(iii),
for
more
details].
Here,
F
Φ
is
a
category.
In
general,
a
Frobenioid
C
is
a
category
equipped
with
a
functor
C
→
F
Φ
to
an
elementary
Frobenioid
F
Φ
satisfying
certain
properties
[cf.
Definition
1.3
for
more
details]
to
the
effect
that
the
structure
of
C
is
“substantially
reflected”
in
this
functor
C
→
F
Φ
.
From
the
point
of
view
of
conventional
arithmetic
geometry,
a
Frobenioid
may
be
thought
of
as
a
sort
of
a
category-theoretic
abstraction
of
the
theory
of
divisors
and
line
bundles
on
models
of
finite
separable
extensions
of
a
given
function
field
or
number
field.
That
is
to
say,
the
base
category
D
corresponds
to
the
category
of
models
of
finite
separable
extensions
of
a
given
function
field
or
number
field;
the
functor
Φ
corresponds
to
the
divisors
on
such
models;
the
“N
≥1
portion”
of
F
Φ
corresponds
to
the
operation
of
multiplying
a
divisor
by
an
element
n
∈
N
≥1
[or,
if
one
considers
the
line
bundle
associated
to
such
a
divisor,
to
the
operation
of
forming
the
n-th
tensor
power
of
the
line
bundle].
In
some
sense,
the
main
result
of
the
theory
of
present
paper
is
the
following:
Under
various
technical
conditions,
the
functor
C
→
F
Φ
that
determines
the
structure
of
C
as
a
Frobenioid
may
be
reconstructed
purely
category-
theoretically,
i.e.,
from
the
structure
of
C
as
a
category
[cf.
Corollary
4.11].
These
technical
conditions
are
typically
satisfied
by
Frobenioids
that
arise
naturally
from
arithmetic
geometry
[cf.
Theorems
6.2,
6.4].
Also,
we
observe
that
these
technical
conditions
appear
unlikely
to
be
superfluous.
Indeed,
we
also
give
various
examples,
involving
Frobenioids
which
do
not
satisfy
various
of
these
technical
conditions,
of
equivalences
of
categories
with
respect
to
which
various
portions
of
THE
GEOMETRY
OF
FROBENIOIDS
I
3
the
functor
C
→
F
Φ
are
not
preserved
[cf.
Examples
3.5,
3.6,
3.7,
3.8,
3.9,
3.10,
4.3].
Perhaps
the
most
fundamental
example
of
this
phenomenon
of
“the
intrinsic
category-theoretic
reconstruction
of
C
→
F
Φ
from
C”
is
the
following.
The
proto-
type
of
a
base
category
D
is
given
by
[the
subcategory
of
connected
objects
of]
a
Galois
category,
i.e.,
a
category
in
which
the
monoids
of
endomorphisms
of
objects
have
the
structure
of
finite
groups.
On
the
other
hand,
the
prototype
of
the
“non-
base
category
portion”
of
a
Frobenioid,
i.e.,
the
“relative
structure
of
C
over
D”,
is
given
by
the
monoid
“F”
[or,
more
generally,
the
monoids
“F
M
”]
discussed
above.
Then
one
central
aspect
of
the
phenomenon
that
“the
relative
structure
of
C
over
D
is
never
confused
with
the
structure
of
D”
is
illustrated
by
the
following
easily
verified
observation:
If
G
is
a
finite
group,
then
any
homomorphism
of
monoids
F
→
G
factors
through
the
natural
surjection
F
N
≥1
.
[We
refer
to
Remark
3.1.2
for
more
details.]
Note
that
this
property
fails
to
hold
if,
for
instance,
one
replaces
F
=
F
Z
≥0
by
Z
≥0
[and
the
surjection
F
N
≥1
by
the
surjection
Z
≥0
{0}].
Put
another
way,
this
property
may
be
thought
of
as
a
consequence
of
the
non-abelian
nature
of
F.
In
particular,
if
one
thinks
of
the
category-theoretic
reconstructibility
of
the
functor
“C
→
F
Φ
”
as
a
sort
of
rigidity,
then
this
property
is
vaguely
reminiscent
of
the
“extraordinary
rigidity”
asserted
by
Grothendieck
in
descriptions
of
his
anabelian
philosophy.
After
defining
and
examining
the
first
properties
of
Frobenioids
in
§1,
we
pro-
ceed
to
discuss,
in
§2,
various
versions
of
“Frobenius
functors”
on
Frobenioids,
which
are
intended
as
category-theoretic
abstractions
of
the
Frobenius
morphism
in
positive
characteristic
algebraic
geometry
[cf.
Remark
6.2.1].
In
§3,
we
begin
the
category-theoretic
reconstruction
of
the
functor
“C
→
F
Φ
”
by
showing
that,
under
certain
conditions,
the
base
category
and
“Frobenius
degree”
[i.e.,
in
effect,
the
“N
≥1
portion
of
F
Φ
”]
may
be
reconstructed
category-theoretically
[cf.
Theorem
3.4].
In
the
theory
of
§3,
we
apply
a
certain
purely
category-theoretic
technique,
which
we
shall
refer
to
as
“slim
exponentiation”;
this
technique
is
entirely
independent
of
the
theory
of
Frobenioids
and
is
discussed
in
detail
in
the
Appendix.
In
§4,
we
then
complete
the
category-theoretic
reconstruction
of
the
functor
“C
→
F
Φ
”
by
show-
ing
that,
under
certain
conditions,
the
divisor
monoid
Φ
may
also
be
reconstructed
category-theoretically
[cf.
Theorem
4.9].
In
§5,
we
study
the
extent
to
which,
under
certain
conditions,
one
may
write
down
“explicit
models”
of
fairly
general
Frobenioids,
in
a
fashion
reminiscent
of
the
explicit
description
of
the
elementary
Frobenioid
F
Φ
[cf.
Theorem
5.2].
This
study
leads
naturally
to
the
investigation
of
various
auxiliary
structures
on
a
Frobenioid,
namely,
base
sections
and
base-
Frobenius
pairs,
that
may
be
used
to
relate
a
given
Frobenioid
satisfying
certain
conditions
to
such
a
“model
Frobenioid”
[cf.
Theorem
5.2,
(iv)].
One
important
technique
in
the
theory
of
§3,
§4,
§5
is
the
operation
of
pass-
ing
from
a
Frobenioid
to
the
perfection
or
realification
of
the
Frobenioid.
Roughly
speaking,
from
the
point
of
view
of
the
monoid
F
=
F
Z
≥0
introduced
above,
these
4
SHINICHI
MOCHIZUKI
operations
correspond,
respectively,
to
passing
from
“Z
≥0
”
to
the
monoids
“Q
≥0
”
[in
the
case
of
the
perfection]
or
“R
≥0
”
[in
the
case
of
the
realification].
Another
important
technique
in
this
theory
is
the
operation
of
passing
to
the
birationaliza-
tion
of
a
Frobenioid.
This
may
be
thought
of
as
a
category-theoretic
abstraction
of
the
notion
of
“working
with
rational
functions”
in
algebraic
geometry;
alternatively,
from
the
point
of
view
of
the
monoid
F
=
F
Z
≥0
introduced
above,
it
may
be
thought
of
as
corresponding
to
the
operation
of
passing
from
“Z
≥0
”
to
the
groupification
“Z”
of
Z
≥0
.
operation
effect
on
Z
≥0
⊆
F
perfection
Z
≥0
Q
≥0
realification
Z
≥0
R
≥0
birationalization
Z
≥0
Z
Finally,
in
§6,
we
consider
the
main
motivating
examples
of
Frobenioids
that
arise
from
number
fields
and
function
fields.
In
particular,
we
observe
in
passing
that
this
“Frobenioid-theoretic
formulation
of
the
elementary
arithmetic
of
number
fields”
also
gives
rise
to
some
interesting
“Frobenioid-theoretic
interpretations”
of
such
classical
results
in
number
theory
as
the
Dirichlet
unit
theorem
and
Tchebotarev’s
density
theorem,
as
well
as
a
result
in
transcendence
theory
due
to
Lang
[cf.
Theo-
rem
6.4,
(i),
(iii),
(iv)].
§I2.
Abstract
Combinatorialization
of
Arithmetic
Geometry
From
a
somewhat
more
conceptual
point
of
view,
one
central
theme
of
the
present
paper
is
the
goal
of
“abstract
combinatorialization
of
scheme-theoretic
arithmetic
geometry”.
Classical
examples
of
this
phenomenon
of
“abstract
com-
binatorialization”
may
be
seen
in
the
theory
of
Galois
categories
or
the
theory
of
monoids
in
the
geometry
of
log
schemes
[or,
more
classically,
toric
varieties].
That
is
to
say,
even
if
one
starts
by
considering
various
finite
étale
coverings
of
schemes,
the
associated
Galois
category
is
a
purely
“abstract
combinatorial”
mathematical
object
that
captures
the
“Galois
structure”
of
the
various
coverings
involved
in
a
fashion
that
is
entirely
independent
of
scheme
theory.
In
a
similar
vein,
although
a
monoid
of
the
sort
that
appears
in
log
geometry
arises
as
a
submonoid
of
the
multiplicative
monoid
determined
by
some
commutative
ring,
the
“abstract
com-
binatorial”
structure
of
such
a
monoid
is
sufficient
to
capture
various
essential
properties
[such
as
normality,
etc.]
of
the
ring
structure
of
the
ambient
ring
in
a
fashion
that
is
entirely
independent
of
ring/scheme
theory.
A
somewhat
less
classical
example
of
this
phenomenon
of
“abstract
combinato-
rialization
of
scheme
theory”
is
given
by
the
theory
of
[Mzk8],
where
it
is
shown
that
very
general
locally
noetherian
log
schemes
may
be
“represented”
by
categories,
in
the
sense
that
equivalences
between
such
categories
arise
from
uniquely
determined
isomorphisms
of
log
schemes.
The
theory
of
[Mzk8]
is
generalized
in
[Mzk9]
so
as
to
take
into
account
the
archimedean
primes
of
log
schemes
which
are
locally
of
finite
THE
GEOMETRY
OF
FROBENIOIDS
I
5
type
over
a
Zariski
localization
of
[the
ring
of
rational
integers]
Z.
As
is
discussed
in
the
introduction
to
[Mzk8],
this
kind
of
result
is
motivated
partly
by
the
an-
abelian
philosophy
of
Grothendieck,
but
perhaps
more
essentially
by
the
idea
that
instead
of
working
with
set-theoretic
objects,
such
as
schemes
or
log
schemes,
one
should
regard
categories
—
which
may
be
thought
of
as
“abstract
combinatorial”
mathematical
objects
constituted
by
some
abstract
collection
of
arrows
—
as
the
“fundamental,
primitive
objects”
of
mathematics
discourse.
Thus,
Grothendieck’s
anabelian
philosophy
may
be
regarded
as
a
“special
case”
of
this
point
of
view,
i.e.,
the
case
where
the
categories
in
question
are
Galois
categories
⎛
abstract,
⎟
⎜
⎜
combinatorial
⎟
⎜
⎟
⎜
⎟
⎜
mathematical
⎟
⎝
⎠
objects
⎛
⎞
⊇
categories
⎛
⎞
⎜
⎟
⎜
—
i.e.,
abstract
⎟
⎜
⎟
⎜
⎟
⎜
collections
⎟
⎠
⎝
⊇
⎞
⎟
⎜
⎜
Galois
⎟
⎟
⎜
⎟
⎜
⎜
categories
⎟
⎝
⎠
of
arrows
—
cf.
the
“absolute
anabelian
geometry”
developed
in
[Mzk5],
[Mzk6],
[Mzk7],
[Mzk10],
[Mzk11],
[Mzk12],
[Mzk14].
One
important
drawback
of
the
“anabelian
branch”
of
this
category-theoretic
approach
to
mathematics
is
that
although
it
is
very
well-suited
to
capturing
essential
aspects
of
the
geometry
of
schemes
at
nonarchimedean
primes,
it
is
ill-suited
to
capturing
the
archimedean
aspects
of
the
geometry
of
schemes,
and,
in
particular,
those
aspects
of
the
global
geometry
of
schemes
over
number
fields
—
such
as
heights
—
that
are
of
interest
in
Diophantine
geometry.
Thus,
from
this
point
of
view,
the
extension
given
in
[Mzk9]
of
the
theory
of
[Mzk8]
has
the
virtue,
relative
to
anabelian
geometry,
of
providing
a
natural
way
to
incorporate
such
archimedean
and
global
phenomena
as
the
global
degree
of
an
arithmetic
line
bundle
over
a
number
field
[cf.
[Mzk9],
Example
5.1]
into
the
above-mentioned
category-theoretic
approach
to
mathematics.
The
approach
of
[Mzk9],
however,
has
the
following
fundamental
drawback:
The
categories
of
[Mzk9]
are
quite
“large”
and
“complicated”
by
comparison
to
Galois
categories,
in
the
sense
that
they
include
a
very
diverse
collection
of
arithmetic
schemes,
by
comparison
to
the
finite
étale
coverings
of
a
fixed
scheme.
This
makes
it
relatively
easy
to
reconstruct
the
original
arithmetic
log
scheme
from
the
category.
On
the
other
hand,
this
relative
ease
of
reconstruction
is
a
reflection,
in
essence,
of
the
fact
that
the
geometry
of
such
categories
is
really
not
so
different
from
the
conventional
geometry
of
arithmetic
log
schemes.
Thus,
in
other
words,
one
doesn’t
gain
very
much
in
the
way
of
essentially
new
geometric
phenomena
by
working
with
such
categories,
relative
to
the
conventional
geometry
of
arithmetic
log
schemes.
By
contrast,
the
relatively
simple
structure
of
Galois
categories
[cf.
also
the
categories
of
[Mzk13]]
makes
it
much
more
difficult
to
reconstruct
the
scheme
from
the
category
—
indeed,
such
a
reconstruction
is
only
possible
in
the
case
of
very
6
SHINICHI
MOCHIZUKI
special
“anabelian”
schemes
—
but,
on
the
other
hand,
this
difficulty
of
reconstruc-
tion
may
be
regarded
as
a
reflection
of
the
fact
that
there
is
indeed
some
interesting
new
geometry
that
arises
from
working
with
Galois
categories
that
does
not
exist
in
the
conventional
geometry
of
schemes.
Perhaps
the
most
fundamental
exam-
ple
of
this
phenomenon
is
the
well-known
fact
that
the
absolute
Galois
groups
of
non-isomorphic
finite
fields
are
isomorphic.
Another
less
elementary
example
of
this
phenomenon
is
the
well-known
fact
that
the
Galois
category
associated
to
a
nonarchimedean
mixed-characteristic
local
field
[i.e.,
a
finite
extension
of
the
p-adic
number
field]
admits
self-equivalences
[i.e.,
the
associated
absolute
Galois
group
admits
automorphisms]
that
do
not
arise
from
scheme
theory
[i.e.,
from
an
isomor-
phism
of
fields
—
cf.,
e.g.,
[NSW],
p.
674].
Put
another
way,
the
difference
between
the
“geometry
of
categories”
—
i.e.,
the
approach
to
arithmetic
geometry
constituted
by
working
with
the
strictly
category-
theoretic
properties
of
categories
—
and
the
classical
approach
to
arithmetic
geom-
etry
constituted
by
working
with
set-theoretic
objects
equipped
with
various
compli-
cated
auxiliary
structures
may
be
regarded
as
analogous
to
the
difference
between
working
with
the
notion
of
an
abstract
group
and
working
with
groups
of
explicit
matrices.
That
is
to
say,
working
with
strictly
group-theoretic
properties
of
abstract
groups
allows
one
to
contemplate
various
structures
that
are
common
to
various
distinct
groups
of
explicit
matrices,
but
which
are
not
so
evident
if
one
happens
to
be
ignorant
of
the
notion
of
an
“abstract
group”
and
hence
obliged
to
restrict
oneself
to
manipulations
involving
explicit
matrices.
This
state
of
affairs
prompts
the
following
question:
Can
one
perhaps
represent
certain
special
arithmetic
log
schemes
of
inter-
est
by
categories
whose
“level
of
complexity”
is
closer
to
Galois
categories
[i.e.,
substantially
simpler
than
the
categories
of
[Mzk9]]
—
thus
allowing
one
to
hope
that
the
geometry
of
such
categories
exhibits
fundamentally
new
phenomena
that
do
not
appear
in
the
conventional
geometry
of
arith-
metic
log
schemes
—
on
the
one
hand,
but
which
nevertheless
allow
one
to
work
naturally
with
archimedean
primes
and
heights
on
the
other?
This
sort
of
question
constituted
one
of
the
principal
motivations
for
the
author
to
develop
the
theory
discussed
in
the
present
paper.
The
answer
to
the
above
question
constituted
by
the
theory
of
present
paper
is,
in
a
word,
the
notion
of
a
Frobenioid.
From
the
point
of
view
of
the
question
posed
above:
Frobenioids
provide
a
single
framework
[cf.
the
notion
of
a
“Galois
cate-
gory”;
the
role
of
monoids
in
log
geometry]
that
allows
one
to
capture
the
essential
aspects
of
both
the
Galois
and
the
divisor
theory
of
number
fields,
on
the
one
hand,
and
function
fields,
on
the
other,
in
such
a
way
that
one
may
continue
to
work
with,
for
instance,
global
degrees
of
arithmetic
line
bundles
on
a
number
field,
but
which
also
exhibits
the
new
phenome-
non
[not
present
in
the
classical
theory
of
number
fields]
of
a
“Frobenius
endomorphism”
of
the
Frobenioid
associated
to
a
number
field.
THE
GEOMETRY
OF
FROBENIOIDS
I
7
Here,
we
remark
that
the
base
category
D
is
typically
a
category
that
is
of
a
level
of
“simplicity”
[cf.
the
above
discussion]
that
is
reminiscent
of
a
Galois
category
[cf.
also
the
“temperoids”
of
[Mzk11];
the
categories
of
Riemann
surfaces
discussed
in
[Mzk13],
§2].
Indeed,
in
the
examples
of
§6,
the
base
category
is
[the
subcategory
of
connected
objects
of]
a
Galois
category.
From
this
point
of
view,
the
main
ingredients
of
a
Frobenioid
—
that
is
to
say,
roughly
speaking,
“Galois”
[i.e.,
the
base
category
D],
“Frobenius”
[i.e.,
“N
≥1
”],
and
“metrics/integral
structures”
[i.e.,
the
family
of
monoids
Φ]
—
are
reminiscent
of
the
theory
of
the
“ring
of
p-adic
periods”
B
crys
of
p-adic
Hodge
theory.
§I3.
Frobenius
Endomorphisms
of
a
Number
Field
From
a
somewhat
less
conceptual
point
of
view,
one
of
the
main
motivations
for
the
author
in
developing
the
theory
of
Frobenioids
came
from
the
long-term
goal
of
developing
a
sort
of
arithmetic
Teichmüller
theory
for
number
fields
equipped
with
an
elliptic
curve,
in
a
fashion
that
is
analogous
to
the
p-adic
Teichmüller
theory
of
[Mzk1],
[Mzk2].
That
is
to
say,
here
one
wishes
to
regard
number
fields
as
corresponding
to
hyperbolic
curves
over
finite
fields
and
elliptic
curves
[over
a
number
field]
as
corresponding
to
the
nilpotent
ordinary
indigenous
bundles
[on
a
hyperbolic
curve
over
a
finite
field]
of
[Mzk1],
[Mzk2].
In
the
p-adic
Teichmüller
theory
of
[Mzk1],
[Mzk2],
certain
canonical
Frobenius
liftings
play
a
central
role.
Thus,
since
Frobenius
liftings
are,
literally,
liftings
of
the
Frobenius
morphism
in
positive
characteristic,
in
order
to
develop
an
“arithmetic
Teichmüller
theory”
for
number
fields
equipped
with
an
elliptic
curve,
one
must
first
have
an
analogue
for
number
fields
of
the
Frobenius
morphism
in
positive
characteristic
scheme
theory.
If
one
starts
to
consider
such
an
analogue
from
a
completely
naive
point
of
view,
then
one
must
contend
with
the
fact
that,
if,
for
instance,
n
≥
2
is
a
integer,
then
the
morphism
p
→
p
n
[where
p
is
a
prime
number]
clearly
does
not
extend
to
a
ring
homomorphism
Z
→
Z!
That
is
to
say,
it
is
difficult
to
see
how
to
accommodate
such
a
“Frobenius
morphism
for
number
fields”
within
the
framework
of
scheme
theory.
On
the
other
hand,
if
one
works
with
monoids
as
in
the
theory
of
log
schemes,
then
such
a
morphism
“p
→
p
n
”
does
indeed
make
sense.
Moreover,
even
if,
for
instance,
one
considers
roots
π
of
p,
the
mapping
π
→
π
n
is
Galois-equivariant.
Thus,
in
summary:
One
important
motivation
for
the
author
in
developing
the
theory
of
Frobenioids
was
the
goal
of
developing
a
geometric
framework
—
i.e.,
roughly
speaking,
a
geometry
built
up
solely
from
“Galois
theory”
and
“monoids”
—
in
which
a
“Frobenius
morphism
on
number
fields”
may
be
constructed.
8
SHINICHI
MOCHIZUKI
Once
one
has
constructed
such
a
“Frobenius
morphism
on
number
fields”,
the
next
step
to
realizing
an
“arithmetic
Teichmüller
theory”
consists
of
construct-
ing
a
“canonical
Frobenius
lifting”.
Although
the
construction
of
such
“canonical
Frobenius
liftings”
lies
[well!]
beyond
the
scope
of
the
present
paper,
we
remark
that
the
ideas
that
lie
behind
such
a
construction
are
motivated
by
the
[scheme-
theoretic!]
Hodge-Arakelov
theory
of
elliptic
curves
surveyed
in
[Mzk3],
[Mzk4],
a
theory
in
which
the
theta
function
on
a
Tate
curve
plays
a
central
role.
In
partic-
ular,
in
order
to
construct
“canonical
Frobenius
liftings”,
it
is
necessary
to
extract
the
essential
“abstract,
combinatorial
content”
of
the
scheme-theoretically
formu-
lated
Hodge-Arakelov
theory
of
[Mzk3],
[Mzk4].
In
fact,
certain
aspects
of
such
an
“extraction
process”
are
achieved
precisely
by
applying
the
theory
of
Frobenioids,
as
is
done
in
a
certain
sequel
to
the
present
paper
and
[Mzk15]
—
namely,
[Mzk16].
Here,
we
pause
to
observe
that
to
pass
from
the
geometry
of
schemes
to,
say,
the
geometry
of
Frobenioids
amounts
to
a
certain
“partial
dismantling
of
scheme
theory”,
i.e.,
to
“forgetting”
a
certain
portion
of
scheme
theory.
As
discussed
above,
one
wants
to
execute
such
a
“partial
dismantling
of
scheme
theory”
precisely
in
order
to
allow
the
construction
of
such
objects
as
a
“Frobenius
morphism
on
number
fields”
which
are
not
possible
within
the
framework
of
scheme
theory.
On
the
other
hand,
if
the
dismantling
process
that
one
executes
is
too
drastic,
then
there
is
a
danger
of
destroying
so
much
of
the
geometry
of
scheme
theory
that
one
is
not
left
with
a
geometry
that
is
sufficiently
rich
so
as
to
allow
the
further
development
of
the
theory.
From
this
point
of
view,
one
of
the
main
themes
of
the
present
paper
[and
[Mzk15]]
consists
of
verifying
that:
The
geometry
of
Frobenioids
retains
a
substantial
portion
of
the
geometry
of
scheme
theory
and,
in
particular,
is
sufficiently
rich
so
as
to
permit
the
execution
of
many
geometric
constructions
and
arguments
familiar
from
scheme
theory.
The
centerpiece
of
this
verification
process
is
the
reconstruction
of
the
functor
“C
→
F
Φ
”,
as
discussed
in
§I1.
Another
aspect
of
this
verification
process,
which
may
be
seen
throughout
the
theory
of
the
present
paper,
is
the
step-by-step
translation
of
various
scheme-theoretic
terms
and
constructions
that
appear
in
the
theory
of
divisors
and
line
bundles
on
models
of
finite
separable
extensions
of
a
given
function
field
or
number
field
into
purely
category-theoretic
language.
For
instance,
one
important
example
of
this
“step-by-step
translation”
is
the
theory
of
base
sections
and
base-Frobenius
pairs
developed
in
§5,
which
may
be
thought
of
as
a
sort
of
category-theoretic
translation
of
the
notion
of
the
tautological
section
of
a
trivial
line
bundle
[cf.
Remark
5.6.1].
§I4.
Étale-like
vs.
Frobenius-like
Structures
Finally,
let
us
return
to
the
“main
result”
discussed
in
§I1,
i.e.,
the
reconstruc-
tion
of
the
functor
“C
→
F
Φ
”.
One
way
to
think
about
this
result
is
that
it
is
a
statement
to
the
effect
that:
THE
GEOMETRY
OF
FROBENIOIDS
I
9
The
structure
of
a
[“permissible”]
base
category
D
[e.g.,
the
subcategory
of
connected
objects
of
a
Galois
category]
is
fundamentally
combina-
torially
different
—
indeed,
different
in
a
category-theoretically
distin-
guishable
fashion
—
from
the
structure
of
the
“Frobenius
portion”
F
of
a
Frobenioid.
This
phenomenon
may
be
thought
of
as
a
sort
of
fundamental
dichotomy
between
types
of
combinatorial
structures
—
i.e.,
between
“étale-like”
structures
which
are
“indifferent
to
order”
[cf.
the
finite
groups
that
as
appear
as
Galois
groups
in
a
Galois
category]
and
“Frobenius-like”
structures
which
are
“order-conscious”
[cf.
the
monoids
“Z
≥0
”,
“N
≥1
”
that
constitute
the
standard
Frobenioid
F].
One
may
also
think
of
“étale-like”
structures
as
“descent-compatible”
structures,
whereas
“Frobenius-like”
structures
are
“descent-incompatible”,
in
the
sense
that
compati-
bility
with
“descent”
may
be
thought
of
as
a
sort
of
violation
of
the
“order”
con-
stituted
by
“the
object
upstairs”
in
the
descent
operation
and
the
“the
object
downstairs”.
Relative
to
the
theme
of
“abstract
combinatorialization”
discussed
in
§I2,
the
point
here
is
that
the
difference
between
“étale-like”
and
“Frobenius-like”
structures
is
an
intrinsic
structural
difference,
not
just
a
matter
of
“arbitrar-
ily
imposed
labels
motivated
by
scheme
theory”
[such
as
“base
category”,
“divisor
monoid”,
“Frobenius
degree”,
etc.]!
For
more
on
this
fundamental
dichotomy
be-
tween
“étale-like”
and
“Frobenius-like”
categorical
structures,
we
refer
to
Remark
3.1.3.
Acknowledgements:
I
would
like
to
thank
Akio
Tamagawa,
Makoto
Matsumoto,
Kazuhiro
Fujiwara,
and
Seidai
Yasuda
for
various
useful
comments.
Also,
I
would
like
to
thank
Yuichiro
Taguchi
for
inviting
me
to
speak
at
Kyushu
University
during
the
Summer
of
2003
on
a
preliminary
version
of
the
theory
discussed
in
this
paper.
10
SHINICHI
MOCHIZUKI
Section
0:
Notations
and
Conventions
Sets:
If
E
is
a
partially
ordered
set,
then
we
shall
denote
by
Order(E)
the
category
whose
objects
are
elements
e
∈
E,
and
whose
morphisms
e
1
→
e
2
[where
e
1
,
e
2
∈
E]
are
the
relations
e
1
≤
e
2
.
Numbers:
We
denote
by
N
≥1
the
[discrete]
multiplicative
monoid
of
rational
integers
≥
1
and
by
Primes
the
set
of
prime
numbers.
Thus,
one
may
think
of
N
≥1
as
the
free
commutative
monoid
generated
by
Primes.
We
shall
write:
def
def
R
>0
=
{a
∈
R
|
a
>
0}
⊆
R
≥0
=
{a
∈
R
|
a
≥
0}
⊆
R
We
shall
refer
to
an
element
Λ
∈
{Z,
Q,
R}
def
def
def
as
a
monoid
type
and
write
Λ
>0
=
Λ
R
>0
⊆
R,
Λ
≥0
=
Λ
R
≥0
⊆
R,
N
=
Z
≥0
.
Also,
we
shall
refer
to
a
monoid
isomorphic
to
[the
additive
monoid]
Λ
≥0
as
a
Λ-
monoprime
monoid
and
to
a
monoid
which
is
a
Λ-monoprime
monoid
for
some
Λ
as
monoprime.
If
M
is
a
Q-monoprime
monoid,
then
we
shall
write
M
⊗
R
≥0
for
the
R-monoprime
monoid
obtained
by
completing
M
relative
to
the
topology
defined
by
the
ordering
on
the
monoid
M
.
We
shall
refer
to
as
a
number
field
any
finite
extension
of
the
field
of
rational
numbers.
Monoids:
Observe
that
any
[not
necessarily
commutative!]
monoid
M
may
be
thought
of
as
a
special
type
of
category,
i.e.,
the
category
with
precisely
one
object
whose
endomorphisms
are
given
by
the
monoid
M
.
THE
GEOMETRY
OF
FROBENIOIDS
I
11
Write
Mon
for
the
category
of
commutative
monoids
[relative
to
some
universe
fixed
throughout
the
discussion].
Let
M
be
an
object
of
Mon;
the
monoid
operation
of
M
will
be
written
additively.
We
shall
denote
by
M
±
⊆
M
the
submonoid
[which,
in
fact,
forms
a
group]
of
invertible
elements
of
M
,
by
M
M
char
=
M/M
±
def
the
quotient
monoid
of
M
by
M
±
,
which
we
shall
refer
to
as
the
characteristic
of
M
,
and
by
M
→
M
gp
the
natural
homomorphism
from
M
to
its
groupification
M
gp
.
Thus,
M
gp
is
the
monoid
[which
is,
in
fact,
a
group]
given
by
the
set
of
equivalence
classes
of
pairs
(a,
b)
∈
M
×
M
,
where
two
such
pairs
(a
1
,
b
1
);
(a
2
,
b
2
)
are
considered
equivalent
if
a
1
+
b
2
+
c
=
b
1
+
a
2
+
c,
for
some
c
∈
M
,
and
the
monoid
operation
on
this
set
is
the
monoid
operation
induced
by
the
monoid
operation
of
M
.
We
shall
say
that
M
is
torsion-free
if
M
has
no
torsion
elements;
we
shall
say
that
M
is
sharp
if
M
±
=
0;
we
shall
say
that
M
is
integral
if
the
natural
map
M
→
M
gp
is
injective;
we
shall
say
that
M
is
saturated
if
every
a
∈
M
gp
for
which
n
·
a
lies
in
the
image
of
M
for
some
n
∈
N
≥1
lies
in
the
image
of
M
.
Denote
by
M
pf
the
perfection
of
M
,
that
is
to
say,
the
inductive
limit
of
the
inductive
system
I
∗
of
monoids
n·
.
.
.
→
M
−→M
→
.
.
.
given
by
assigning
to
each
element
of
a
∈
N
≥1
a
copy
of
M
,
which
we
denote
by
I
a
,
and
to
every
two
elements
a,
b
∈
M
such
that
a
divides
b
the
morphism
def
I
a
=
M
→
I
b
=
M
given
by
multiplication
by
n
=
b/a.
Thus,
the
object
I
1
of
the
inductive
system
I
∗
determines
a
natural
morphism
M
→
M
pf
which
is
injective
if
M
is
torsion-free,
integral,
and
saturated,
hence,
in
particular,
if
M
is
sharp,
integral,
and
saturated.
We
shall
say
that
M
is
perfect
if
multiplication
by
any
element
of
N
≥1
on
M
is
bijective.
Thus,
M
pf
is
always
perfect;
M
is
perfect
if
and
only
if
the
natural
map
M
→
M
pf
is
an
isomorphism.
Note
that
M
is
saturated
if
and
only
if
M
char
is.
We
shall
say
that
M
is
of
characteristic
type
if
the
fibers
of
the
natural
map
M
→
M
char
are
torsors
over
M
±
.
Note
that
if
M
is
of
characteristic
type,
then
M
is
integral
if
and
only
if
M
char
is.
If
φ
:
M
1
→
M
2
is
a
morphism
of
Mon,
then
we
shall
say
that
φ
is
characteristically
injective
if
φ
is
injective,
and,
moreover,
the
morphism
M
1
char
→
M
2
char
induced
by
φ
is
injective.
12
SHINICHI
MOCHIZUKI
Now
suppose
that
M
is
sharp,
integral,
and
saturated.
If
a,
b
∈
M
,
then
we
shall
write
a
≤
b
if
∃
c
∈
M
such
that
a
+
c
=
b
and
a
b
if
∃n
∈
N
≥1
such
that
a
≤
n
·
b.
If
a
subset
S
⊆
M
satisfies
the
property
that
there
exists
a
b
∈
M
such
that
a
≤
b
for
all
a
∈
S,
then
we
shall
say
that
S
is
bounded
[by
b].
If
S
⊆
M
is
a
subset
and
b
∈
M
,
then
we
shall
write
def
Bound
S
(b)
=
{a
∈
S
|
a
≤
b}
[i.e.,
Bound
S
(b)
is
the
maximal
subset
of
S
that
is
bounded
by
b].
Observe
that
if
M
is
R-monoprime,
then
every
bounded
subset
S
⊆
M
possesses
a
[unique]
supremum
sup(S)
∈
M
[i.e.,
S
is
bounded
by
b
if
and
only
if
b
≥
sup(S)].
We
shall
say
that
0
=
a
∈
M
is
irreducible
if
any
equation
a
=
b
+
c
in
M
,
where
b,
c
∈
M
,
implies
that
b
=
0
or
c
=
0.
We
shall
say
that
0
=
a
∈
M
is
primary
if
for
any
M
b
a,
where
b
=
0,
it
holds
that
a
b.
Denote
by
Primary(M
)
the
set
of
primary
elements
of
M
.
One
verifies
immediately
that
the
relation
“a
b”
[where
a,
b
∈
Primary(M
)]
determines
an
equivalence
relation
on
Primary(M
).
A
-equivalence
class
of
elements
of
Primary(M
)
will
be
referred
to
as
a
prime
of
M
.
[Note
that
this
notion
of
a
“prime”
differs
from
the
conventional
notion
of
a
“prime
ideal”
of
M
.]
Denote
by
Prime(M
)
the
set
of
primes
of
M
.
If
p
∈
Prime(M
),
then
we
shall
denote
by
M
p
⊆
M
the
submonoid
generated
by
the
elements
contained
in
the
subset
p
⊆
M
.
Note
that
each
subset
p
⊆
M
,
where
p
∈
Prime(M
),
is
closed
under
multiplication
by
elements
of
N
≥1
,
and
that
Primary(M
pf
)
=
{a
∈
M
pf
|
∃n
∈
N
≥1
such
that
n
·
a
∈
Primary(M
)}
Primary(M
)
=
Primary(M
pf
)
M
∼
Prime(M
)
→
Prime(M
pf
)
[where
we
regard
M
as
a
subset
of
M
pf
via
the
natural
inclusion].
Finally,
we
observe
that
the
relation
“≤”
on
elements
of
M
determines
a
category
Order(M
)
[via
the
partially
ordered
set
structure
on
M
determined
by
“≤”].
THE
GEOMETRY
OF
FROBENIOIDS
I
13
Topological
Groups:
Let
G
be
a
Hausdorff
topological
group,
and
H
⊆
G
a
closed
subgroup.
Let
us
write
def
Z
G
(H)
=
{g
∈
G
|
g
·
h
=
h
·
g,
∀
h
∈
H}
for
the
centralizer
of
H
in
G.
If
Π
is
a
profinite
group,
then
we
shall
write
B(Π)
for
the
category
whose
objects
are
finite
sets
equipped
with
a
continuous
Π-action
and
whose
morphisms
are
morphisms
of
Π-sets.
Thus,
B(Π)
is
a
Galois
category,
or,
in
the
terminology
of
[Mzk7],
a
connected
anabelioid.
If
Z
Π
(H)
=
{1}
for
every
open
subgroup
H
⊆
Π,
then
we
shall
say
that
Π
is
slim.
Categories:
Let
C
be
a
category.
We
shall
denote
the
collection
of
objects
(respectively,
arrows)
of
C
by:
Ob(C)
(respectively,
Arr(C))
The
opposite
category
to
C
will
be
denoted
by
C
opp
.
A
category
with
precisely
one
object
will
be
referred
to
as
a
one-object
category;
a
category
with
precisely
one
morphism
[which
is
necessarily
the
identity
morphism
of
the
unique
object
of
such
a
category]
will
be
referred
to
as
a
one-morphism
category.
Thus,
a
one-morphism
category
is
always
a
one-object
category.
If
A
∈
Ob(C)
is
an
object
of
C,
then
we
shall
denote
by
C
A
the
category
whose
objects
are
morphisms
B
→
A
of
C
and
whose
morphisms
[from
an
object
B
1
→
A
to
an
object
B
2
→
A]
are
A-morphisms
B
1
→
B
2
in
C
and
by
A
C
the
category
whose
objects
are
morphisms
A
→
B
of
C
and
whose
morphisms
(from
an
object
A
→
B
1
to
an
object
A
→
B
2
)
are
morphisms
B
1
→
B
2
in
C
that
are
compatible
with
the
given
arrows
A
→
B
1
,
A
→
B
2
.
Thus,
we
have
a
natural
functor
(j
A
)
!
:
C
A
→
C
[given
by
forgetting
the
structure
morphism
to
A].
Similarly,
if
f
:
A
→
B
is
a
morphism
in
C,
then
f
defines
a
natural
functor
f
!
:
C
A
→
C
B
14
SHINICHI
MOCHIZUKI
by
mapping
an
arrow
[i.e.,
an
object
of
C
A
]
C
→
A
to
the
object
of
C
B
given
by
the
composite
C
→
A
→
B
with
f
.
We
shall
call
an
object
A
∈
Ob(C)
terminal
(respectively,
pseudo-terminal)
if
for
every
object
B
∈
Ob(C),
there
exists
a
unique
arrow
(respectively,
there
exists
a
[not
necessarily
unique!]
arrow)
B
→
A
in
C.
We
shall
say
that
two
arrows
of
a
category
are
co-objective
if
their
domains
and
codomains
coincide.
We
shall
say
that
an
arrow
β
:
B
→
A
of
a
category
C
is
fiberwise-surjective
if,
for
every
arrow
γ
:
C
→
A
of
C,
there
exist
arrows
δ
B
:
D
→
B,
δ
C
:
D
→
C
such
that
β
◦
δ
B
=
γ
◦
δ
C
.
An
arrow
of
a
category
which
is
a
fiberwise-surjective
monomorphism
will
be
referred
to
as
an
FSM-morphism.
One
verifies
immediately
that
every
composite
of
FSM-morphisms
is
again
an
FSM-morphism.
A
category
C
which
satisfies
the
property
that
every
FSM-morphism
of
C
is,
in
fact,
an
isomor-
phism
will
be
referred
to
as
a
category
of
FSM-type.
Let
C
be
a
category;
A
∈
Ob(C).
Write
End
C
(A);
Aut
C
(A)
for
the
monoids
of
endomorphisms
and
automorphisms
of
A
in
C,
respectively.
We
shall
say
that
an
endomorphism
α
∈
End
C
(A)
of
C
is
a
sub-automorphism
if
there
exists
an
arrow
φ
:
B
→
A
of
C
and
an
automorphism
β
∈
Aut
C
(B)
such
that
φ
◦
β
=
α
◦
φ;
write
(Aut
C
(A)
⊆)
Aut
sub
C
(A)
⊆
End
C
(A)
for
the
subset
of
End
C
(A)
determined
by
the
sub-automorphisms
of
A.
We
shall
say
that
A
is
Aut-saturated
(respectively,
Aut
sub
-saturated;
of
Aut-type)
if
Aut
C
(A)
=
sub
Aut
sub
C
(A)
(respectively,
Aut
C
(A)
=
End
C
(A);
Aut
C
(A)
=
End
C
(A)).
If
every
object
of
C
is
Aut-saturated
(respectively,
Aut
sub
-saturated;
of
Aut-type),
then
we
shall
say
that
C
is
Aut-saturated
(respectively,
Aut
sub
-saturated;
of
Aut-type).
We
shall
say
that
an
arrow
A
→
B
of
C
is
an
End-equivalence
if
there
exists
an
arrow
B
→
A
in
C.
We
shall
refer
to
a
natural
transformation
between
functors
all
of
whose
com-
ponent
morphisms
are
isomorphisms
as
an
isomorphism
between
the
functors
in
question.
If
φ
:
C
1
→
C
2
is
a
functor
between
categories
C
1
,
C
2
,
then
we
shall
denote
by
Aut(φ)
—
or,
when
there
is
no
fear
of
confusion,
Aut(C
1
→
C
2
)
—
the
group
of
automorphisms
of
the
functor
φ,
and
by
End(φ)
—
or,
when
there
is
no
fear
of
confusion,
End(C
1
→
C
2
)
—
the
monoid
of
natural
transformations
from
the
functor
φ
to
itself.
We
shall
say
that
φ
is
rigid
if
Aut(φ)
is
trivial.
A
category
C
will
be
called
slim
if
the
natural
functor
C
A
→
C
is
rigid,
for
every
A
∈
Ob(C).
We
recall
that
if
Π
is
a
profinite
THE
GEOMETRY
OF
FROBENIOIDS
I
15
group,
then
Π
is
slim
if
and
only
if
the
category
B(Π)
is
slim
[cf.
[Mzk7],
Corollary
1.1.6].
A
diagram
of
functors
between
categories
will
be
called
1-commutative
if
the
various
composite
functors
in
question
are
isomorphic.
When
such
a
diagram
“com-
mutes
in
the
literal
sense”
we
shall
say
that
it
0-commutes.
Note
that
when
a
dia-
gram
in
which
the
various
composite
functors
are
all
rigid
“1-commutes”,
it
follows
from
the
rigidity
hypothesis
that
any
isomorphism
between
the
composite
functors
in
question
is
necessarily
unique.
Thus,
to
state
that
such
a
diagram
1-commutes
does
not
result
in
any
“loss
of
information”
by
comparison
to
the
datum
of
a
specific
isomorphism
between
the
various
composites
in
question.
A
category
C
will
be
called
a
skeleton
if
any
two
isomorphic
objects
of
C
are,
in
fact,
equal.
A
skeletal
subcategory
of
a
category
C
is
a
full
subcategory
S
⊆
C
such
that
S
is
a
skeleton,
and,
moreover,
the
inclusion
functor
S
→
C
is
an
equivalence
of
categories.
We
shall
say
that
a
nonempty
[i.e.,
non-initial]
object
A
∈
Ob(C)
is
connected
if
it
is
not
isomorphic
to
the
coproduct
of
two
nonempty
objects
of
C.
We
shall
say
that
an
object
A
∈
Ob(C)
is
mobile
if
there
exists
an
object
B
∈
Ob(C)
such
that
the
set
Hom
C
(A,
B)
has
cardinality
≥
2
[i.e.,
the
diagonal
from
this
set
to
the
prod-
uct
of
this
set
with
itself
is
not
bijective].
We
shall
say
that
an
object
A
∈
Ob(C)
is
quasi-connected
if
it
is
either
immobile
[i.e.,
not
mobile]
or
connected.
Thus,
con-
nected
objects
are
always
quasi-connected.
We
shall
say
that
a
category
C
is
totally
(respectively,
almost
totally)
epimorphic
if
every
morphism
in
C
whose
domain
is
arbitrary
(respectively,
nonempty)
and
whose
codomain
is
arbitrary
(respectively,
connected)
is
an
epimorphism.
We
shall
say
that
C
is
of
finitely
(respectively,
countably)
connected
type
if
it
is
closed
under
formation
of
finite
(respectively,
countable)
coproducts;
every
object
of
C
is
a
coproduct
of
a
finite
(respectively,
countable)
collection
of
connected
objects;
and,
moreover,
all
finite
(respectively,
countable)
coproducts
A
i
in
the
category
satisfy
the
condition
that
the
natural
map
Hom
C
(B,
A
i
)
→
Hom
C
(B,
A
i
)
is
bijective,
for
all
connected
B
∈
Ob(C).
If
C
is
of
finitely
or
countably
connected
type,
then
every
nonempty
object
of
C
is
mobile;
in
particular,
a
nonempty
object
of
C
is
connected
if
and
only
if
it
is
quasi-connected.
If
a
mobile
object
A
∈
Ob(C)
satisfies
the
condition
that
every
morphism
in
C
whose
domain
is
nonempty
and
whose
codomain
is
A
is
an
epimorphism,
then
A
∼
is
connected.
[Indeed,
C
1
C
2
→
A,
where
C
1
,
C
2
are
nonempty,
implies
that
the
composite
map
Hom
C
(A,
B)
→
Hom
C
(A,
B)
×
Hom
C
(A,
B)
→
Hom
C
(C
1
,
B)
×
Hom
C
(C
2
,
B)
=
Hom
C
(C
1
∼
C
2
,
B)
→
Hom
C
(A,
B)
is
bijective,
for
all
B
∈
Ob(C).]
In
particular,
it
follows
that
if
C
is
a
totally
epimorphic
category,
then
every
object
of
C
is
quasi-connected.
16
SHINICHI
MOCHIZUKI
If
C
is
a
category
of
finitely
or
countably
connected
type,
then
we
shall
write
C
0
⊆
C
for
the
full
subcategory
of
connected
objects.
[Note,
however,
that
in
general,
objects
of
C
0
are
not
necessarily
connected
—
or
even
quasi-connected
—
as
objects
of
C
0
!]
On
the
other
hand,
if,
in
addition,
C
is
almost
totally
epimorphic,
then
C
0
is
totally
epimorphic
[so
every
object
of
C
is
quasi-connected].
If
C
is
a
category,
then
we
shall
write
C
⊥
(respectively,
C
)
for
the
category
formed
from
C
by
taking
arbitrary
“formal”
[possibly
empty]
finite
(respectively,
countable)
coproducts
of
objects
in
C.
That
is
to
say,
we
define
the
“Hom”
of
C
⊥
(respectively,
C
)
by
the
following
formula:
Hom(
def
A
i
,
i
B
j
)
=
j
Hom
C
(A
i
,
B
j
)
i
j
[where
the
A
i
,
B
j
are
objects
of
C].
Thus,
C
⊥
(respectively,
C
)
is
a
category
of
finitely
(respectively,
countably)
connected
type.
Note
that
objects
of
C
define
connected
objects
of
C
⊥
or
C
.
Moreover,
there
are
natural
[up
to
isomorphism]
equivalences
of
categories
∼
(C
⊥
)
0
→
C;
∼
(C
)
0
→
C;
∼
(D
0
)
⊥
→
D;
∼
(E
0
)
→
E
for
D
(respectively,
E)
a
category
of
finitely
connected
type
(respectively,
category
of
countably
connected
type).
If
C
is
a
totally
epimorphic
category,
then
C
⊥
(re-
spectively,
C
)
is
an
almost
totally
epimorphic
category
of
finitely
(respectively,
countably)
connected
type.
In
particular,
the
operations
“0”,
“⊥”
(respectively,
“”)
define
one-to-one
correspondences
[up
to
equivalence]
between
the
totally
epimorphic
categories
and
the
almost
totally
epimorphic
categories
of
finitely
(respectively,
countably)
con-
nected
type.
We
observe
in
passing
that
if
C
is
a
totally
epimorphic
category,
and
α
◦
β
[where
α,
β
∈
Arr(C)]
is
an
isomorphism,
then
α,
β
are
isomorphisms.
If
C
is
a
[small]
category,
then
we
shall
write
G(C)
for
the
graph
associated
to
C.
This
graph
is
the
graph
with
precisely
one
vertex
for
each
object
of
C
and
precisely
one
edge
for
each
arrow
of
C
[joining
the
vertices
corresponding
to
the
domain
and
codomain
of
the
arrow].
We
shall
refer
to
the
full
subcategories
of
C
determined
by
the
objects
and
arrows
that
compose
a
connected
component
of
the
graph
G(C)
as
a
connected
component
of
C.
In
particular,
we
shall
say
that
C
is
connected
if
G(C)
is
connected.
[Note
that
by
working
with
respect
to
some
“sufficiently
large”
enveloping
universe,
it
makes
sense
to
speak
of
a
category
which
is
not
necessarily
small
at
being
connected.]
THE
GEOMETRY
OF
FROBENIOIDS
I
17
Given
two
arrows
f
i
:
A
i
→
B
i
(where
i
=
1,
2)
in
a
category
C,
we
shall
refer
to
a
commutative
diagram
∼
A
1
→
A
2
⏐
⏐
⏐
f
⏐
f
1
B
1
2
∼
→
B
2
—
where
the
horizontal
arrows
are
isomorphisms
in
C
—
as
an
abstract
equivalence
from
f
1
to
f
2
.
If
there
exists
an
abstract
equivalence
from
f
1
to
f
2
,
then
we
shall
say
that
f
1
,
f
2
are
abstractly
equivalent.
If
C
1
,
C
2
,
and
D
are
categories,
and
Φ
1
:
C
1
→
D;
Φ
2
:
C
2
→
D
are
functors,
then
we
define
the
“CFP”
—
i.e.,
“categorical
fiber
product”
—
C
1
×
D
C
2
of
C
1
,
C
2
over
D
to
be
the
category
whose
objects
are
triples
∼
(A
1
,
A
2
,
α
:
Φ
1
(A
1
)
→
Φ
2
(A
2
))
where
A
i
∈
Ob(C
i
)
(for
i
=
1,
2);
α
is
an
isomorphism
of
D;
and
whose
morphisms
∼
∼
(A
1
,
A
2
,
α
:
Φ
1
(A
1
)
→
Φ
2
(A
2
))
→
(B
1
,
B
2
,
β
:
Φ
1
(B
1
)
→
Φ
2
(B
2
))
are
pairs
of
morphisms
γ
i
:
A
i
→
B
i
[in
C
i
,
for
i
=
1,
2]
such
that
β
◦
Φ
1
(γ
1
)
=
Φ
2
(γ
2
)◦α.
One
verifies
easily
that
if
Φ
2
is
an
equivalence,
then
the
natural
projection
functor
C
1
×
D
C
2
→
C
1
is
also
an
equivalence.
Let
C
be
a
category;
S
a
collection
of
arrows
in
C;
φ
∈
Arr(C).
Then
we
shall
say
that
φ
is
minimal-adjoint
to
S
(respectively,
minimal-coadjoint
to
S;
mid-adjoint
to
S)
if
every
factorization
φ
=
α
◦
β
(respectively,
φ
=
β
◦
α;
φ
=
α
◦
β
◦
γ)
of
φ
in
C
such
that
β
lies
in
S
satisfies
the
property
that
β
is,
in
fact,
an
isomorphism.
If
φ
admits
a
factorization
φ
=
α
◦
β
◦
γ
in
C,
then
we
shall
say
that
β
is
subordinate
to
φ.
If
φ
is
not
an
isomorphism,
but,
for
every
factorization
φ
=
α
◦
β
in
C,
it
holds
that
either
α
or
β
is
an
isomorphism,
then
we
shall
say
that
φ
is
irreducible.
We
shall
refer
to
an
FSM-morphism
which
is
irreducible
as
an
FSMI-morphism.
Thus,
a
category
of
FSM-type
does
not
contain
any
FSMI-morphisms.
We
shall
say
that
a
category
C
is
of
FSMFF-type
[i.e.,
“FSM-finitely
factorizable
type”]
if
the
following
two
conditions
hold:
(a)
every
FSM-morphism
of
C
which
is
not
an
isomorphism
factors
as
a
composite
of
finitely
many
FSMI-morphisms;
(b)
for
every
A
∈
Ob(C),
there
exists
a
natural
number
N
such
that
for
every
composite
φ
n
◦
φ
n−1
◦
.
.
.
◦
φ
2
◦
φ
1
18
SHINICHI
MOCHIZUKI
of
FSMI-morphisms
φ
1
,
.
.
.
,
φ
n
such
that
the
domain
of
φ
1
is
equal
to
A,
it
holds
that
n
≤
N
.
Thus,
if
C
is
of
FSM-type,
then
it
is
of
FSMFF-type.
Also,
we
observe
that
[by
condition
(b)]
no
endomorphism
of
an
object
of
a
category
of
FSMFF-type
is
an
FSMI-morphism.
If
C
is
a
totally
epimorphic
category,
A
∈
Ob(C),
and
G
⊆
Aut
C
(A)
is
a
subgroup,
then
we
shall
say
that
an
arrow
φ
:
A
→
B
of
C
is
a
categorical
quotient
of
A
by
G
if
the
following
conditions
hold:
(a)
φ
◦
γ
=
φ,
for
all
γ
∈
G;
(b)
for
every
morphism
ψ
:
A
→
C
such
that
ψ
◦
γ
=
ψ
for
all
γ
∈
G,
there
exists
a
unique
morphism
ψ
:
B
→
C
such
that
ψ
=
ψ
◦
φ.
If
φ
:
A
→
B
is
a
categorical
quotient
of
A
by
G,
then
we
shall
say
that
A
→
B
is
mono-minimal
if
the
following
condition
holds:
For
every
factorization
φ
=
φ
◦
ζ,
where
ζ
:
A
→
A
is
a
monomorphism
such
that
there
exists
a
subgroup
G
⊆
Aut
C
(A
),
together
with
an
isomorphism
∼
G
→
G
that
is
compatible,
relative
to
ζ,
with
the
respective
actions
of
G,
G
on
A,
A
[which
implies,
by
total
epimorphicity,
that
φ
:
A
→
B
is
a
categorical
quotient
of
A
by
G
],
it
holds
that
ζ
is
an
isomorphism.
Thus,
[by
total
epimorphicity]
it
follows
that
an
isomorphism
is
always
a
mono-minimal
categorical
quotient
of
its
domain
by
the
trivial
group.
If
C
is
a
category,
then
we
shall
say
that
A
∈
Ob(C)
is
an
anchor
if
there
only
exist
finitely
many
isomorphism
classes
of
objects
of
A
C
that
arise
from
irreducible
arrows
A
→
B.
We
shall
say
that
A
∈
Ob(C)
is
a
subanchor
if
there
exists
an
arrow
A
→
B,
where
B
is
an
anchor.
If
C
is
a
totally
epimorphic
category,
then
we
shall
say
that
A
∈
Ob(C)
is
an
iso-subanchor
if
there
exist
a
subanchor
B
∈
Ob(C),
a
subgroup
G
⊆
Aut
C
(B),
and
a
morphism
B
→
A
[in
C]
which
is
a
mono-minimal
categorical
quotient
of
B
by
G.
THE
GEOMETRY
OF
FROBENIOIDS
I
19
Section
1:
Definitions
and
First
Properties
In
the
present
§1,
we
discuss
the
notion
of
a
Frobenioid,
which
may
be
thought
of
as
a
category
whose
internal
structure
behaves
roughly
like
that
of
an
“elemen-
tary
Frobenioid”.
An
“elementary
Frobenioid”
is,
in
essence,
a
sort
of
semi-direct
product
of
the
multiplicative
monoid
N
≥1
[which
is
to
be
thought
of
as
a
“Frobenius
action”]
with
a
system
of
monoids
[which
are
roughly
of
the
sort
that
appear
in
the
theory
of
log
structures]
on
a
“base
category”
[a
category
which
behaves
roughly
like
a
Galois
category].
We
begin
by
introducing
the
fundamental
notions
of
“elementary
Frobenioids”
and
“pre-Frobenioids”.
Definition
1.1.
(i)
We
shall
say
that
M
∈
Ob(Mon)
is
pre-divisorial
if
it
is
integral
[cf.
§0],
saturated
[cf.
§0],
and
of
characteristic
type
[cf.
§0].
Suppose
that
M
is
pre-
divisorial.
Then
we
shall
say
that
M
is
group-like
if
M
char
is
zero;
we
shall
say
that
M
is
divisorial
if
M
is
sharp
[cf.
§0].
[Thus,
if
M
is
pre-divisorial,
then
M
char
is
divisorial.]
If
α
is
an
endomorphism
of
a
pre-divisorial
monoid
M
∈
Ob(Mon),
then
we
shall
say
that
α
is
non-dilating
if
the
endomorphism
α
char
of
M
char
induced
by
α
is
the
identity
endomorphism
of
M
char
whenever
α
char
(a)
a
for
all
primary
[cf.
§0]
a
∈
M
char
.
(ii)
Let
D
be
a
category.
Then
we
shall
refer
to
a
contravariant
functor
Φ
:
D
→
Mon
as
a
monoid
on
D
if
the
following
conditions
are
satisfied:
(a)
every
morphism
of
monoids
α
∗
:
Φ(A)
→
Φ(B)
induced
by
a
morphism
α
:
B
→
A
of
D
is
char-
acteristically
injective
[cf.
§0];
(b)
if
α
is
an
FSM-morphism
[cf.
§0]
of
D,
then
α
∗
:
Φ(A)
→
Φ(B)
is
an
isomorphism
of
monoids.
If,
moreover,
every
monoid
Φ(A)
[as
A
ranges
over
the
objects
of
D]
(respectively,
some
monoid
Φ(A)
[where
A
∈
Ob(D)])
satisfies
some
property
of
monoids
[e.g.,
is
pre-divisorial,
sharp,
etc.],
then
we
shall
say
that
Φ
(respectively,
A)
satisfies
this
property.
Note
that
if
Φ
is
a
monoid
on
D,
then
Φ
determines
monoids
“Φ
char
”,
“Φ
gp
”,
Φ
pf
”
on
D
[i.e.,
by
assigning
A
→
Φ(A)
char
,
A
→
Φ(A)
gp
,
A
→
Φ(A)
pf
],
which
we
shall
refer
to,
respectively,
as
the
characteristic,
groupification,
and
perfection
of
Φ.
If
Φ
is
pre-
divisorial,
then
we
shall
say
that
Φ
is
non-dilating
if
the
endomorphisms
of
Φ(A),
where
A
∈
Ob(D),
induced
by
endomorphisms
∈
End
D
(A)
are
non-dilating.
(iii)
Let
Φ
be
a
monoid
on
a
category
D.
Then
we
shall
refer
to
as
the
elemen-
tary
Frobenioid
associated
to
Φ
the
category
F
Φ
20
SHINICHI
MOCHIZUKI
defined
as
follows:
The
objects
of
F
Φ
are
the
objects
of
D.
If
A,
B
∈
Ob(F
Φ
),
whose
respective
images
in
D
we
denote
by
A
D
,
B
D
∈
Ob(D),
then
a
morphism
φ
:
A
→
B
of
F
Φ
is
defined
to
be
a
collection
of
data
(φ
D
,
Z
φ
,
n
φ
)
where
φ
D
:
A
D
→
B
D
is
a
morphism
of
D;
Z
φ
∈
Φ(A
D
);
n
φ
∈
N
≥1
.
Here,
φ
D
(respectively,
A
D
)
will
be
referred
to
as
the
projection
Base(φ)
(respectively,
Base(A))
of
φ
(respectively,
A)
to
D;
Z
φ
as
the
zero
divisor
Div(φ)
of
φ;
and
n
φ
as
def
the
Frobenius
degree
deg
Fr
(φ)
of
φ.
If
C
D
=
Base(C)
∈
Ob(D),
then
the
composite
of
two
morphisms
φ
=
(φ
D
,
Z
φ
,
n
φ
)
:
A
→
B;
ψ
=
(ψ
D
,
Z
ψ
,
n
ψ
)
:
B
→
C
is
given
as
follows:
ψ
◦
φ
=
(ψ
D
◦
φ
D
,
φ
∗D
(Z
ψ
)
+
n
ψ
·
Z
φ
,
n
ψ
·
n
φ
)
:
A
→
C
Observe
that
the
assignment
Φ
→
F
Φ
is
functorial
with
respect
to
homomorphisms
of
functors
[on
D]
valued
in
monoids
Φ
→
Φ
;
also,
we
have
a
natural
projection
functor:
F
Φ
→
D
We
shall
refer
to
the
D
as
the
base
category
of
F
Φ
.
If
M
∈
Ob(Mon),
then
observe
that
the
elementary
Frobenioid
F
Φ
M
associated
to
the
functor
Φ
M
on
any
one-
morphism
[cf.
§0]
category
that
assigns
to
the
unique
object
of
the
category
the
monoid
M
is
itself
a
one-object
[cf.
§0]
category,
whose
endomorphism
monoid
we
shall
denote
by
F
M
and
refer
to
as
the
elementary
Frobenioid
associated
to
M
.
[Thus,
the
notation
“F
”
denotes
a
category
(respectively,
monoid)
when
the
subscript
“”
is
a
functor
(respectively,
monoid).]
More
explicitly,
the
underlying
set
of
F
M
is
the
product
M
×
N
≥1
equipped
with
the
monoid
structure
is
given
as
follows:
if
a
1
,
a
2
∈
M
,
n
1
,
n
2
∈
N
≥1
,
def
then
(a
1
,
n
1
)
·
(a
2
,
n
2
)
=
(a
1
+
n
1
·
a
2
,
n
1
·
n
2
).
Also,
we
shall
write
F
=
F
Z
≥0
and
refer
to
F
as
the
standard
Frobenioid.
(iv)
Let
D,
Φ,
F
Φ
be
as
in
(iii);
C
a
category.
Assume
further
that
Φ
is
divisorial,
and
that
C,
D
are
connected,
totally
epimorphic
categories
[cf.
§0].
Then
we
shall
refer
to
a
[covariant]
functor
C
→
F
Φ
as
a
pre-Frobenioid
structure
on
C.
The
natural
projection
functor
F
Φ
→
D
thus
restricts
to
a
natural
projection
functor
C→D
on
C;
similarly,
the
operations
“Base(−)”,
“Div(−)”,
“deg
Fr
(−)”
on
F
Φ
restrict
to
operations
on
C
which
[by
abuse
of
notation]
we
shall
denote
by
the
same
notation.
THE
GEOMETRY
OF
FROBENIOIDS
I
21
We
shall
refer
to
the
D
as
the
base
category
of
C.
By
abuse
of
notation,
we
shall
often
regard
Φ
as
a
functor
on
C
[i.e.,
by
composing
the
original
functor
Φ
with
the
natural
projection
functor
C
→
D]
and
apply
similar
terminology
to
objects
of
C
and
“Φ
as
a
functor
on
C”
to
the
terminology
applied
to
objects
of
D
and
“Φ
as
a
functor
on
D”
[cf.
(ii)].
We
shall
refer
to
a
category
C
equipped
with
a
pre-Frobenioid
structure
C
→
F
Φ
as
a
pre-Frobenioid
and
to
the
monoid
Φ
as
the
divisor
monoid
of
the
pre-Frobenioid.
Remark
1.1.1.
If
φ
◦
ψ
is
a
composite
of
morphisms
φ,
ψ
of
a
pre-Frobenioid,
then
the
operations
“Base(−)”,
“Div(−)”,
“deg
Fr
(−)”
behave
in
the
following
way
under
composition:
Base(φ
◦
ψ)
=
Base(φ)
◦
Base(ψ)
Div(φ
◦
ψ)
=
(Base(ψ))
∗
(Div(φ))
+
deg
Fr
(φ)
·
Div(ψ)
deg
Fr
(φ
◦
ψ)
=
deg
Fr
(φ)
·
deg
Fr
(ψ)
Indeed,
this
follows
immediately
from
the
definition
of
an
elementary
Frobenioid
in
Definition
1.1,
(iii).
Next,
we
introduce
various
terms
to
describe
types
of
morphisms
and
objects
in
a
pre-Frobenioid.
Definition
1.2.
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
pre-Frobenioid;
φ
∈
Arr(C).
Write
φ
:
A
→
B
[where
def
def
A,
B
∈
Ob(C)];
A
D
=
Base(A)
∈
Ob(D),
B
D
=
Base(B)
∈
Ob(D).
Then:
(i)
We
shall
say
that
φ
is
linear
if
deg
Fr
(φ)
=
1.
We
shall
say
that
φ
is
isometric,
or,
alternatively,
an
isometry,
if
Div(φ)
=
0
[cf.
Definition
1.1,
(iii)].
If
ψ
∈
Arr(C)
is
co-objective
with
φ
[cf.
§0],
then
we
shall
say
that
φ,
ψ
are
metrically
equivalent
if
Div(φ)
=
Div(ψ).
(ii)
We
shall
refer
to
φ
as
a
base-isomorphism
(respectively,
base-FSM-morphism)
if
Base(φ)
is
an
isomorphism
(respectively,
FSM-morphism
[cf.
§0])
in
D.
We
shall
refer
to
two
objects
of
C
that
map
to
isomorphic
objects
of
D
as
base-isomorphic.
We
shall
refer
to
φ
as
a
pull-back
morphism
if
the
natural
transformation
of
con-
travariant
functors
on
C
Hom
C
(−,
A)
→
Hom
C
(−,
B)
×
Hom
D
(−,B
D
)|
C
(Hom
D
(−,
A
D
)|
C
)
[where
“|
C
”
denotes
the
restriction
of
a
functor
on
D
to
a
functor
on
C
via
the
natural
projection
functor
C
→
D]
induced
by
φ
is
an
isomorphism.
If
ψ
∈
Arr(C)
is
co-
objective
with
φ
[cf.
§0],
then
we
shall
say
that
φ,
ψ
are
base-equivalent
(respectively,
Div-equivalent)
if
Base(φ)
=
Base(ψ)
(respectively,
Φ(φ)
=
Φ(ψ)).
If
A
=
B
[i.e.,
φ
is
an
endomorphism],
then
we
shall
say
that
φ
is
a
base-identity
(respectively,
22
SHINICHI
MOCHIZUKI
Div-identity)
endomorphism
if
it
is
base-equivalent
(respectively,
Div-equivalent)
to
the
identity
endomorphism
of
A.
Write
O
×
(A)
⊆
Aut
C
(A);
O
(A)
⊆
End
C
(A)
for
the
submonoids
of
base-identity
linear
endomorphisms.
(iii)
We
shall
say
that
φ
is
a
pre-step
[a
term
motivated
by
the
point
of
view
that
the
only
possibly
non-isomorphic
portion
of
such
a
morphism
is
the
“step”
constituted
by
a
non-zero
zero
divisor]
if
it
is
a
linear
base-isomorphism.
If
φ
is
a
pre-step,
then
we
shall
say
that
it
is
a
step
(respectively,
a
primary
pre-step)
if
φ
is
not
an
isomorphism
(respectively,
if
the
zero
divisor
Div(φ)
∈
Φ(A)
of
φ
is
a
primary
[cf.
§0]
element
of
the
monoid
Φ(A)).
We
shall
say
that
φ
is
co-angular
[a
term
that
arises
from
a
certain
“coincidence
of
angles”
that
occurs
for
co-angular
morphisms
in
the
case
of
Frobenioids
that
arise
in
an
archimedean
context
—
cf.
[Mzk15],
Definition
3.1,
(iii)]
if,
for
any
factorization
φ
=
α
◦
β
◦
γ
in
C,
where
α
is
linear,
β
is
an
isometric
pre-step,
and
either
α
or
γ
is
a
base-isomorphism,
it
follows
that
β
is
an
isomorphism.
We
shall
say
that
φ
is
LB-invertible
[i.e.,
“line
bundle-
invertible”
—
a
term
motivated
by
the
isomorphism
induced
by
such
a
morphism
between
the
“image
line
bundle
of
the
domain”
and
“the
line
bundle
portion
of
the
codomain”
in
the
case
of
various
Frobenioids
that
arise
from
arithmetic
geometry]
if
it
is
co-angular
and
isometric.
We
shall
say
that
φ
is
a
morphism
of
Frobenius
type
[a
term
motivated
by
the
fact
that,
in
the
case
of
Frobenioids
that
arise
from
arithmetic
geometry,
such
a
morphism
corresponds
to
simply
“raising
to
the
n-th
tensor
power”
for
some
n
∈
N
≥1
]
if
φ
is
an
LB-invertible
base-isomorphism.
We
shall
say
that
φ
is
a
prime-Frobenius
morphism,
or,
alternatively,
a
deg
Fr
(φ)-Frobenius
morphism,
if
it
is
a
morphism
of
Frobenius
type
such
that
deg
Fr
(φ)
∈
Primes
[cf.
§0].
(iv)
A
Frobenius-ample
object
of
C
is
defined
to
be
an
object
C
such
that
for
any
n
∈
N
≥1
,
C
admits
an
endomorphism
of
Frobenius
degree
n.
A
Frobenius-trivial
object
of
C
is
defined
to
be
an
object
C
such
that
there
exists
a
homomorphism
of
monoids
ζ
:
N
≥1
→
End
C
(C)
which
satisfies
the
following
properties:
(a)
the
composite
of
ζ
with
the
map
to
N
≥1
given
by
the
Frobenius
degree
is
the
identity
on
N
≥1
;
(b)
the
endomorphisms
in
the
image
of
ζ
are
base-identity
endomorphisms
of
Frobenius
type.
A
Div-Frobenius-trivial
object
of
C
is
defined
to
be
an
object
C
such
that
there
exists
a
homomorphism
of
monoids
ζ
:
N
≥1
→
End
C
(C)
which
satisfies
the
following
properties:
(a)
the
composite
of
ζ
with
the
map
to
N
≥1
given
by
the
Frobenius
degree
is
the
identity
on
N
≥1
;
(b)
the
endomorphisms
in
the
image
of
ζ
are
Div-identity
endomorphisms
of
Frobenius
type.
A
universally
Div-Frobenius-trivial
object
of
C
is
defined
to
be
an
object
C
such
that
for
every
pull-back
morphism
C
→
C
of
C,
it
follows
that
C
is
a
Div-Frobenius-trivial
object.
A
quasi-Frobenius-trivial
object
of
C
is
defined
to
be
an
object
C
such
that
for
any
n
∈
N
≥1
,
C
admits
a
base-identity
endomorphism
[which
is
not
necessarily
of
Frobenius
type!]
of
Frobenius
degree
n.
A
sub-quasi-Frobenius-trivial
object
of
C
is
defined
to
be
an
object
C
such
that
there
exists
a
co-angular
pre-step
D
→
C
in
C
such
that
D
is
quasi-Frobenius
trivial.
A
metrically
trivial
object
of
C
is
defined
to
be
an
object
C
such
that
for
any
co-angular
pre-step
C
→
D,
it
holds
that
THE
GEOMETRY
OF
FROBENIOIDS
I
23
D
is
isomorphic
to
C.
A
base-trivial
object
of
C
is
defined
to
be
an
object
C
such
that
any
object
D
∈
Ob(C)
such
that
Base(C)
∼
=
Base(D)
[in
D]
is,
in
fact,
sub
isomorphic
to
C.
An
Aut-ample
(respectively,
Aut
-ample;
End-ample)
object
def
of
C
is
defined
to
be
an
object
C
such
that,
if
we
write
C
D
=
Base(C),
then
sub
the
natural
map
Aut
C
(C)
→
Aut
D
(C
D
)
(respectively,
Aut
sub
C
(C)
→
Aut
D
(C
D
);
End
C
(C)
→
End
D
(C
D
))
is
surjective.
A
perfect
object
of
C
is
defined
to
be
an
object
C
such
that
for
every
n
∈
N
≥1
,
it
holds
that
every
B
∈
Ob(C)
base-isomorphic
to
C
appears
as
the
codomain
of
a
morphism
of
Frobenius
type
of
Frobenius
degree
n,
and,
moreover,
for
every
pair
of
morphisms
of
Frobenius
type
φ
1
:
B
1
→
B
1
,
φ
2
:
B
2
→
B
2
of
Frobenius
degree
n,
where
B
1
,
B
2
are
base-isomorphic
to
C,
and
every
pre-step
ψ
:
B
1
→
B
2
,
there
exists
a
unique
pre-step
ψ
:
B
1
→
B
2
such
that
ψ
◦
φ
1
=
φ
2
◦
ψ.
A
group-like
object
of
C
is
defined
to
be
an
object
C
such
that
Φ(C)
=
0
[or,
equivalently,
Φ(C)
is
group-like
—
cf.
the
conventions
of
Definition
1.1,
(i),
(ii),
(iv)].
A
Frobenius-compact
object
of
C
is
defined
to
be
an
object
C
such
that
O
×
(C)
is
commutative,
O
×
(C)
pf
=
0,
and
every
element
of
Aut
C
(C)
that
acts
on
O
×
(C)
pf
via
multiplication
by
an
element
∈
Q
>0
in
fact
acts
trivially
on
O
×
(C)
pf
.
A
Frobenius-normalized
object
of
C
is
defined
to
be
an
object
C
such
that
if
φ
∈
End
C
(C)
is
a
base-identity
endomorphism
of
Frobenius
degree
d
∈
N
≥1
,
and
α
∈
O
(C),
then
α
d
◦
φ
=
φ
◦
α.
A
unit-trivial
object
of
C
is
defined
to
be
an
object
C
such
that
O
×
(C)
=
{1}.
An
isotropic
object
[a
term
motivated
by
the
archimedean
case
—
cf.
[Mzk15],
Definition
3.1,
(iii)]
of
C
is
defined
to
be
an
object
C
such
that
any
isometric
pre-step
C
→
D
in
C
is,
in
fact,
an
isomorphism.
We
shall
write
C
istr
⊆
C
for
the
full
subcategory
of
isotropic
objects
and
C
lin
⊆
C;
C
bs-iso
⊆
C;
C
pl-bk
⊆
C
for
the
subcategories
determined,
respectively,
by
the
linear
morphisms,
base-isomor-
phisms,
and
pull-back
morphisms.
We
shall
say
that
φ
:
A
→
B
is
an
isotropic
hull
[of
A]
if
φ
is
an
isometric
pre-step,
B
is
isotropic,
and
for
every
morphism
γ
:
A
→
C,
where
C
is
isotropic,
there
exists
a
unique
morphism
β
:
B
→
C
such
that
γ
=
β
◦φ.
A
Frobenius-isotropic
object
of
C
is
defined
to
be
an
object
C
such
that
there
exists
a
morphism
of
Frobenius
type
C
→
D
such
that
D
is
isotropic.
(v)
If
every
object
of
C
is
Frobenius-ample
(respectively,
Frobenius-trivial;
Div-
Frobenius-trivial;
universally
Div-Frobenius-trivial;
quasi-Frobenius-trivial;
sub-
quasi-Frobenius-trivial;
metrically
trivial;
base-trivial;
Aut-ample;
Aut
sub
-ample;
End-ample;
perfect;
group-like;
Frobenius-compact;
Frobenius-normalized;
unit-
trivial;
isotropic;
Frobenius-isotropic),
then
we
shall
say
that
the
pre-Frobenioid
C
→
F
Φ
is
of
Frobenius-ample
type
(respectively,
of
Frobenius-trivial
type;
of
Div-
Frobenius-trivial
type;
of
universally
Div-Frobenius-trivial
type;
of
quasi-Frobenius-
trivial
type;
of
sub-quasi-Frobenius-trivial
type;
of
metrically
trivial
type;
of
base-
trivial
type;
of
Aut-ample
type;
of
Aut
sub
-ample
type;
of
End-ample
type;
of
perfect
type;
of
group-like
type;
of
Frobenius-compact
type;
of
Frobenius-normalized
type;
of
unit-trivial
type;
of
isotropic
type;
of
Frobenius-isotropic
type).
24
SHINICHI
MOCHIZUKI
Remark
1.2.1.
The
following
implications
follow
formally
from
the
definitions:
pull-back
morphism
which
is
a
base-isomorphism
⇐⇒
isomorphism
base-trivial
=⇒
metrically
trivial
base-identity
=⇒
Div-identity
universally
Div-Frobenius-trivial
=⇒
Div-Frobenius-trivial
We
are
now
ready
to
define
the
notion
of
a
“Frobenioid”.
Definition
1.3.
Let
D,
Φ,
C
→
F
Φ
be
as
in
Definition
1.2.
Then
we
shall
say
that
the
pre-Frobenioid
C
→
F
Φ
[i.e.,
C
equipped
with
this
functor]
is
a
Frobenioid
if
the
following
conditions
are
satisfied:
(i)
(Surjectivity
to
the
Base
Category
via
Pull-back
Morphisms)
(a)
Every
iso-
morphism
class
of
D
arises
as
the
image
via
the
natural
projection
functor
C
→
D
of
an
isomorphism
class
of
a
Frobenius-trivial
object
of
C.
(b)
If
A,
B
∈
Ob(C),
def
∼
def
A
D
=
Base(A),
B
D
=
Base(B),
and
α
:
A
D
→
B
D
is
an
isomorphism,
then
there
exist
pre-steps
φ
:
C
→
A,
ψ
:
C
→
B
such
that
α
=
Base(ψ)
◦
Base(φ)
−1
.
(c)
For
every
A
∈
Ob(C),
the
fully
faithful
[cf.
the
isomorphism
of
functors
appearing
in
the
definition
of
a
“pull-back
morphism”
given
in
Definition
1.2,
(ii)]
functor
def
pl-bk
=
(C
pl-bk
)
A
→
D
A
D
C
A
def
[where
A
D
=
Base(A)]
determined
by
the
natural
projection
functor
C
→
D
is
an
equivalence
of
categories
[cf.
§0].
(ii)
(Surjectivity
to
N
≥1
via
Morphisms
of
Frobenius
Type)
For
every
A
∈
Ob(C),
n
∈
N
≥1
,
there
exists
a
morphism
of
Frobenius
type
φ
:
A
→
B
in
C
of
Frobenius
degree
n;
moreover,
if
ψ
:
A
→
C
is
any
other
morphism
of
Frobenius
type
in
C
of
Frobenius
degree
n,
then
there
exists
a(n)
[unique
—
since
C
is
totally
∼
epimorphic]
isomorphism
β
:
B
→
C
such
that
β
◦
φ
=
ψ.
(iii)
(Surjectivity
to
the
Divisor
Monoid
via
Co-angular
Morphisms)
(a)
The
co-angular
morphisms
of
C
are
closed
under
composition.
(b)
If
A
→
A
is
a
co-
angular
pre-step
of
C,
then
any
morphism
A
→
A
is
co-angular.
(c)
Given
any
co-angular
pre-step
φ
:
A
→
B,
there
exists
a
[uniquely
determined]
bijection
of
monoids
∼
O
(A)
→
O
(B)
such
that
O
(A)
α
→
β
∈
O
(B)
implies
β
◦
φ
=
φ
◦
α;
moreover,
this
bijection
depends
only
[among
the
bijections
induced
by
the
various
co-angular
pre-steps
A
→
B]
on
Base(φ).
(d)
Denote
by
C
coa-pre
⊆
C
the
subcategory
determined
by
the
co-angular
pre-steps.
Then
the
natural
functors
A
C
coa-pre
def
=
A
(C
coa-pre
)
→
Order(Φ(A));
def
coa-pre
C
A
=
(C
coa-pre
)
A
→
Order(Φ(A))
opp
THE
GEOMETRY
OF
FROBENIOIDS
I
25
[obtained
by
assigning
to
an
arrow
φ
:
A
→
B
the
element
Div(φ)
∈
Φ(A)
and
to
∼
an
arrow
ψ
:
B
→
A
the
element
(ψ
∗
)
−1
(Div(ψ))
∈
Φ(A)
[since
ψ
∗
:
Φ(A)
→
Φ(B)
is
a
bijection
—
cf.
the
fact
that
ψ
is
a
base-isomorphism!]
are
equivalences
of
categories.
(iv)
(Factorization
of
Arbitrary
Morphisms)
Let
φ
:
A
→
B
be
a
morphism
of
C.
Then:
(a)
φ
admits
a
factorization
φ
=
α
◦
β
◦
γ
where
α
is
an
pull-back
morphism,
β
is
a
pre-step,
and
γ
is
a
morphism
of
Frobenius
type;
this
factorization
is
unique,
up
to
replacing
the
triple
(α,
β,
γ)
by
a
triple
of
the
form
(α
◦
δ,
δ
−1
◦
β
◦
,
−1
◦
γ),
where
δ,
are
isomorphisms
of
C.
(b)
Every
pull-back
morphism
of
C
is
LB-invertible
and
linear.
(v)
(Factorization
of
Pre-steps)
Let
φ
:
A
→
B
be
a
pre-step
of
C.
Then:
(a)
φ
is
a
monomorphism.
(b)
φ
admits
a
factorization
φ
=
α
◦
β
where
α
is
an
isometric
pre-step,
and
β
is
a
co-angular
pre-step;
this
factorization
is
unique,
up
to
replacing
the
pair
(α,
β)
by
a
pair
of
the
form
(α
◦
γ,
γ
−1
◦
β),
where
γ
is
an
isomorphism
of
C.
(c)
φ
admits
a
factorization
φ
=
α
◦
β
,
where
α
is
a
co-angular
pre-step,
and
β
is
an
isometric
pre-step;
this
factorization
is
unique,
up
to
replacing
the
pair
(α
,
β
)
by
a
pair
of
the
form
(α
◦
γ
,
(γ
)
−1
◦
β
),
where
γ
is
an
isomorphism
of
C.
(vi)
(Faithfulness
up
to
Units)
Let
φ,
ψ
:
A
→
B
be
base-equivalent,
metrically
equivalent
co-angular
pre-steps
of
C.
Then
there
exists
a
[necessarily
unique]
α
∈
O
×
(B)
such
that
φ
=
α
◦
ψ.
(vii)
(Isotropic
Objects)
(a)
For
every
A
∈
Ob(C),
there
exists
a
[necessarily
unique,
up
to
unique
isomorphism]
isotropic
hull
A
→
B.
(b)
If
A
∈
Ob(C)
is
isotropic,
and
A
→
C
is
a
morphism
of
C,
then
C
is
also
isotropic.
Remark
1.3.1.
Note
that
it
follows
from
Definition
1.3,
(iii),
(b),
(c),
that
if
C
is
a
Frobenioid,
then
the
monoid
O
(A)
is
commutative,
for
all
A
∈
Ob(C).
Proposition
1.4.
(Co-angular
and
LB-invertible
Morphisms)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
pre-Frobenioid;
φ
:
A
→
B
a
morphism
of
C.
Then:
(i)
Suppose
that
the
codomain
of
any
arrow
of
C
whose
domain
is
equal
to
A
is
isotropic.
Then
φ
is
co-angular.
In
particular,
φ
is
a
morphism
of
Frobenius
type
if
and
only
if
it
is
an
isometric
base-isomorphism.
(ii)
Suppose
that
C
is
a
Frobenioid.
Then
φ
is
a
pull-back
morphism
if
and
only
if
it
is
an
LB-invertible
linear
morphism
[i.e.,
a
co-angular
linear
isometry].
26
SHINICHI
MOCHIZUKI
(iii)
Suppose
that
C
is
a
Frobenioid.
Then
every
LB-invertible
pre-step
is
an
isomorphism.
(iv)
Suppose
that
C
is
a
Frobenioid.
Then
a
morphism
φ
of
C
is
co-angular
if
and
only
if,
in
the
factorization
φ
=
α
◦
β
◦
γ
of
Definition
1.3,
(iv),
(a),
the
pre-step
β
is
co-angular.
(v)
Suppose
that
C
is
a
Frobenioid.
Then
a
morphism
φ
of
C
is
LB-invertible
if
and
only
if
it
is
of
the
form
α
◦
β,
where
α
is
a
pull-back
morphism,
and
β
is
a
morphism
of
Frobenius
type.
Proof.
Assertion
(i)
follows
formally
from
the
definitions
of
the
terms
“isotropic”,
“isometric
pre-step”,
“co-angular”,
and
“morphism
of
Frobenius
type”
[cf.
Defini-
tion
1.2,
(i),
(iii),
(iv)].
As
for
assertion
(ii),
if
φ
is
a
pull-back
morphism,
then
it
follows
from
Definition
1.3,
(iv),
(b),
that
φ
is
an
LB-invertible
linear
morphism.
Now
suppose
that
φ
is
LB-invertible
and
linear.
Then
by
applying
Remark
1.1.1
to
the
factorization
of
Definition
1.3,
(iv),
(a),
the
fact
that
φ
is
a
linear
isometry
implies
that
φ
may
be
written
in
the
form
α
◦
β,
where
α
is
a
pull-back
morphism,
and
β
is
an
isometric
pre-step.
On
the
other
hand,
since
φ
is
co-angular,
it
follows
that
β
is
an
isomorphism,
hence
that
φ
is
a
pull-back
morphism,
as
desired.
As-
sertion
(iii)
follows
from
either
the
uniqueness
of
the
factorization
of
pre-steps
of
Definition
1.3,
(v),
(b),
or
the
essential
uniqueness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Definition
1.3,
(ii)].
Next,
we
consider
assertion
(iv).
If
β
is
co-angular,
then
since
α,
γ
are
co-
angular
[cf.
assertion
(ii);
Definition
1.2,
(iii)],
it
follows
from
Definition
1.3,
(iii),
(a),
that
φ
is
co-angular.
Conversely,
if
φ
is
co-angular,
and
β
=
β
1
◦
β
2
◦
β
3
,
where
β
2
is
an
isometric
pre-step,
then
by
applying
Remark
1.1.1,
together
with
the
fact
that
D
is
totally
epimorphic
[cf.
the
discussion
of
§0]
to
this
factorization
of
β,
we
conclude
that
β
1
,
β
3
are
pre-steps,
hence
that
α
◦
β
1
is
linear,
and
that
β
3
◦
γ
is
a
base-isomorphism;
thus,
the
co-angularity
of
φ
=
(α
◦
β
1
)
◦
β
2
◦
(β
3
◦
γ)
implies
that
β
2
is
an
isomorphism,
hence
that
β
is
co-angular,
as
desired.
Finally,
we
consider
assertion
(v).
If
φ
=
α◦β,
where
α
is
a
pull-back
morphism,
and
β
is
a
morphism
of
Frobenius
type,
then
[since
α,
β
are
LB-invertible
—
cf.
assertion
(ii);
Definition
1.2,
(iii)]
it
follows
from
Remark
1.1.1
that
φ
is
isometric
and
from
Definition
1.3,
(iii),
(a),
that
φ
is
co-angular,
hence
LB-invertible.
Now
suppose
that
φ
is
LB-invertible,
and
that
we
have
a
factorization
φ
=
α◦β
◦γ,
where
α,
β,
and
γ
are
as
in
Definition
1.3,
(iv),
(a).
By
assertion
(iv),
β
is
co-angular;
by
Remark
1.1.1,
β
is
isometric.
Thus,
β
is
an
LB-invertible
pre-step,
hence
[cf.
assertion
(iii)]
an
isomorphism,
as
desired.
This
completes
the
proof
of
assertion
(v).
Remark
1.4.1.
We
refer
to
the
Chart
of
Types
of
Morphisms
in
a
Frobenioid
given
at
the
end
of
the
present
paper
for
a
summary
of
the
properties
of
the
base
category
projections,
zero
divisors,
and
Frobenius
degrees
satisfied
by
various
types
of
morphisms
in
a
Frobenioid.
THE
GEOMETRY
OF
FROBENIOIDS
I
27
Proposition
1.5.
(Elementary
Frobenioids
are
Frobenioids)
Let
Φ
be
a
pre-divisorial
monoid
on
a
connected,
totally
epimorphic
category
D.
Then:
(i)
F
Φ
,
equipped
with
the
natural
functor
F
Φ
→
F
Φ
char
,
is
a
Frobenioid
of
Aut-
ample,
Aut
sub
-ample,
End-ample,
base-trivial,
Frobenius-trivial,
Frobenius-
normalized,
and
isotropic
type.
(ii)
There
is
a
natural,
functorial
isomorphism
∼
O
(A)
→
Φ(A)
∼
[so
O
×
(A)
→
Φ(A)
±
]
for
objects
A
∈
Ob(F
Φ
).
(iii)
If
all
of
the
monoids
in
the
image
of
Φ
are
perfect
(respectively,
group-
like),
then
F
Φ
is
of
perfect
(respectively,
group-like)
type.
Proof.
Since
D
is
a
connected,
totally
epimorphic
category,
the
fact
that
F
Φ
is
as
well
follows
immediately
from
the
definition
of
the
morphisms
of
F
Φ
in
Defini-
tion
1.1,
(iii);
the
fact
that
a
pre-divisorial
monoid
is
integral
[cf.
Definition
1.1,
(i)];
and
the
injectivity
condition
of
Definition
1.1,
(ii),
(a).
Thus,
F
Φ
is
a
pre-
Frobenioid.
It
is
immediate
from
the
definitions
that
assertion
(ii)
holds,
and
that
all
objects
of
F
Φ
are
Aut-ample,
Aut
sub
-ample,
End-ample,
base-trivial,
Frobenius-
trivial,
Frobenius-normalized,
and
isotropic.
Also,
one
verifies
immediately
[cf.
the
definition
of
the
category
F
Φ
in
Definition
1.1,
(iii)]
that
a
morphism
of
F
Φ
is
a
pull-back
morphism
if
and
only
if
it
is
a
linear
isometry.
The
fact
that
F
Φ
satisfies
the
conditions
of
Definition
1.3
now
follows
immediately
from
the
definition
of
the
category
F
Φ
in
Definition
1.1,
(iii),
together
with
assertion
(ii)
and
the
“explicit
description”
of
co-angular
morphisms
and
morphisms
of
Frobenius
type
in
Propo-
sition
1.4,
(i)
[which
is
applicable
to
all
morphisms
of
F
Φ
since
F
Φ
is
of
isotropic
type].
This
completes
the
proof
of
assertion
(i).
Assertion
(iii)
is
immediate
from
the
definitions
and
assertion
(i).
One
important
technique
for
constructing
new
Frobenioids
is
given
by
the
fol-
lowing
result.
Proposition
1.6.
(Categorical
Fiber
Products)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobenioid.
Let
D
be
a
connected,
totally
epimorphic
category;
D
→
D
a
functor
that
maps
FSM-morphisms
to
FSM-morphisms.
Denote
by
Φ
:
D
→
Mon
the
divisorial
monoid
obtained
by
restricting
Φ
to
D
.
Then:
(i)
There
is
a
natural
equivalence
of
categories
∼
F
Φ
→
F
Φ
×
D
D
[where
the
latter
category
is
the
categorical
fiber
product
of
§0].
28
SHINICHI
MOCHIZUKI
(ii)
The
categorical
fiber
product
[cf.
§0]
def
C
=
C
×
D
D
equipped
with
the
functor
C
→
F
Φ
[obtained
by
applying
“(−)×
D
D
”
to
the
functor
C
→
F
Φ
]
is
a
Frobenioid.
(iii)
A
morphism
of
C
is
a(n)
isometry
(respectively,
morphism
of
a
given
Frobenius
degree;
co-angular
morphism;
LB-invertible
morphism;
pull-
back
morphism)
if
and
only
if
its
projection
to
C
is.
(iv)
A
base-isomorphism
of
C
is
a
morphism
of
Frobenius
type
(respec-
tively,
pre-step;
step)
if
and
only
if
its
projection
to
C
is.
Moreover,
the
projection
∼
functor
C
→
C
determines
a
bijection
of
monoids
O
(A
)
→
O
(A),
for
every
A
∈
Ob(C
)
that
projects
to
A
∈
Ob(C).
(v)
A
object
of
C
is
Frobenius-trivial
(respectively,
quasi-Frobenius-trivial;
sub-quasi-Frobenius-trivial;
metrically
trivial;
base-trivial;
perfect;
group-
like;
unit-trivial;
Frobenius-normalized;
isotropic;
Frobenius-isotropic)
if
and
only
if
it
projects
to
such
an
object
of
C.
(vi)
A
object
of
C
is
Aut-ample
(respectively,
Aut
sub
-ample;
End-ample)
if
it
projects
to
such
an
object
of
C.
Proof.
Assertion
(i)
follows
formally
from
the
definitions.
Next,
observe
that
the
fact
that
D
is
a
totally
epimorphic
category
implies
immediately
that
C
is
as
well;
similarly,
[in
light
of
the
various
properties
of
the
natural
projection
functor
C
→
D
assumed
in
Definition
1.3,
(i),
(a),
(b),
(c)]
the
fact
that
D
is
connected
implies
immediately
that
C
is
also
connected.
Thus,
C
[equipped
with
the
functor
C
→
F
Φ
obtained
by
applying
“(−)
×
D
D
”
to
the
functor
C
→
F
Φ
]
is
a
pre-
Frobenioid.
Now
assertion
(vi)
follows
immediately
from
the
definitions;
one
checks
immediately
that
the
equivalences
of
assertions
(iii),
(iv),
(v)
hold.
In
light
of
these
equivalences,
the
conditions
of
Definition
1.3
follow
via
a
routine
verification.
Thus,
C
is
a
Frobenioid.
This
completes
the
proof
of
assertion
(ii).
Proposition
1.7.
(Composites
of
Morphisms)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobenioid.
Then:
(i)
The
following
classes
of
morphisms
are
closed
under
composition:
isome-
tries,
base-isomorphisms,
base-FSM-morphisms,
pull-back
morphisms,
linear
morphisms,
pre-steps,
co-angular
morphisms,
LB-invertible
mor-
phisms,
morphisms
of
Frobenius
type.
(ii)
A
morphism
of
C
is
a
pull-back
morphism
if
and
only
if
it
is
minimal-
adjoint
to
the
base-isomorphisms
of
C.
A
morphism
of
C
is
a
base-isomorphism
if
and
only
if
it
is
minimal-coadjoint
to
the
pull-back
morphisms
of
C;
alter-
natively,
a
morphism
of
C
is
a
base-isomorphism
if
and
only
if
it
is
may
be
written
THE
GEOMETRY
OF
FROBENIOIDS
I
29
as
a
composite
α
◦
β,
where
α
is
a
pre-step,
and
β
is
a
morphism
of
Frobenius
type.
(iii)
A
morphism
of
C
is
of
Frobenius
type
if
and
only
if
it
is
minimal-
coadjoint
to
the
linear
morphisms
of
C.
A
morphism
of
C
is
linear
if
and
only
if
it
is
minimal-adjoint
to
the
morphisms
of
Frobenius
type
of
C;
alternatively,
a
morphism
of
C
is
linear
if
and
only
if
it
is
may
be
written
as
a
composite
α
◦
β,
where
α
is
a
pull-back
morphism,
and
β
is
a
pre-step.
(iv)
A
pre-step
of
C
is
co-angular
if
and
only
if
it
is
mid-adjoint
[cf.
§0]
to
the
isometric
pre-steps.
(v)
If
a
composite
morphism
φ
=
α◦β
of
C
is
a(n)
isomorphism
(respectively,
base-isomorphism;
linear
morphism;
pre-step;
isometry;
co-angular
pre-
step;
co-angular
linear
morphism;
pull-back
morphism),
then
so
are
α,
β.
If,
moreover,
the
domain
of
φ
is
isotropic,
then
a
similar
statement
holds
for
morphisms
of
Frobenius
type.
Proof.
Assertion
(i)
follows
immediately
from
the
definitions
for
isometries,
base-
isomorphisms,
base-FSM-morphisms,
pull-back
morphisms,
linear
morphisms,
and
pre-steps;
from
Definition
1.3,
(iii),
(a),
for
co-angular
morphisms,
hence
also
for
LB-invertible
morphisms
and
morphisms
of
Frobenius
type.
Next,
the
sufficiency
of
the
various
conditions
given
in
assertions
(ii),
(iii)
follows
immediately
from
[defini-
tions
and]
the
[existence
of
the]
factorization
of
Definition
1.3,
(iv),
(a).
Moreover,
in
light
of
the
existence
of
this
factorization,
the
necessity
of
the
various
conditions
given
in
assertions
(ii),
(iii)
follows
immediately
for
pull-back
morphisms
and
mor-
phisms
of
Frobenius
type
from
the
essential
uniqueness
of
this
factorization
[and
the
total
epimorphicity
of
C];
for
base-isomorphisms
from
the
total
epimorphicity
of
D;
and
for
linear
morphisms
from
the
well-known
structure
of
the
multiplicative
monoid
N
≥1
and
the
essential
uniqueness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Definition
1.3,
(ii)].
In
light
of
Remark
1.1.1,
assertion
(v)
follows
for
isomorphisms
(respectively,
base-isomorphisms;
linear
morphisms;
pre-steps;
isometries)
immediately
from
the
fact
that
C
is
totally
epimorphic
(respectively,
from
the
fact
that
D
is
totally
epi-
morphic;
from
the
well-known
structure
of
the
multiplicative
monoid
N
≥1
;
from
assertion
(v)
for
base-isomorphisms
and
linear
morphisms;
from
the
fact
that
the
monoid
Φ
on
D
is
sharp
[cf.
Definition
1.1,
(i)],
together
with
the
characteristic
in-
jectivity
assumption
of
Definition
1.1,
(ii),
(a)).
Now
assertion
(iv)
follows
formally
from
[the
definitions
and]
assertion
(v)
for
pre-steps
[cf.
the
argument
applied
in
the
proof
of
Proposition
1.4,
(iv)!];
assertion
(v)
for
co-angular
pre-steps
follows
from
assertion
(v)
for
pre-steps
and
assertion
(iv).
To
prove
assertion
(v)
for
co-
angular
linear
morphisms,
suppose
that
φ
is
co-angular
and
linear.
Then
observe
that
by
assertion
(v)
for
linear
morphisms,
α,
β
are
linear.
Thus,
by
applying
the
factorization
for
linear
morphisms
of
assertion
(iii),
together
with
the
factorization
of
Definition
1.3,
(v),
(c)
[cf.
also
Proposition
1.4,
(ii);
assertion
(i)
for
co-angular
linear
morphisms],
we
may
write
α
=
α
1
◦
α
2
,
β
=
β
1
◦
β
2
,
α
2
◦
β
1
=
γ
1
◦
γ
2
,
where
α
1
,
β
1
,
γ
1
are
co-angular
linear
morphisms,
and
α
2
,
β
2
,
γ
2
are
isometric
pre-steps.
30
SHINICHI
MOCHIZUKI
Thus,
φ
=
(α
1
◦
γ
1
)
◦
(γ
2
◦
β
2
),
which
[by
the
co-angularity
of
φ]
implies
that
γ
2
◦
β
2
is
an
isomorphism,
hence
[by
assertion
(v)
for
isomorphisms]
that
β
2
,
γ
2
are
isomor-
phisms.
Thus,
by
the
co-angularity
of
α
2
◦
β
1
=
γ
1
◦
γ
2
,
we
conclude
that
α
2
is
an
isomorphism.
In
particular,
it
follows
that
α,
β
are
co-angular
linear
morphisms,
as
desired.
Now
assertion
(v)
for
pull-back
morphisms
follows
from
assertion
(v)
for
co-angular
linear
isometries
[cf.
also
Proposition
1.4,
(ii)].
Finally,
assertion
(v)
for
morphisms
of
Frobenius
type
in
C
istr
[cf.
Definition
1.3,
(vii),
(b)]
follows
from
assertion
(v)
for
isometric
base-isomorphisms,
since
morphisms
of
C
istr
are
always
co-angular
[cf.
Proposition
1.4,
(i)].
This
completes
the
proof
of
assertion
(v).
Proposition
1.8.
(Pre-steps)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobenioid.
Then:
(i)
If
the
natural
projection
functor
C
→
D
is
full,
then
every
pre-step
of
C
is
a
linear
End-equivalence.
If
D
is
of
Aut-type
[cf.
§0],
then
every
linear
End-equivalence
of
C
is
a
pre-step.
(ii)
Suppose
further
that
C
is
of
metrically
trivial
and
Aut-ample
type.
Then
a
morphism
of
C
is
a
co-angular
pre-step
if
and
only
if
it
is
abstractly
equivalent
[cf.
§0]
to
a
base-identity
pre-step
endomorphism
of
C.
(iii)
An
object
A
∈
Ob(C)
is
non-group-like
if
and
only
if
there
exists
a
co-angular
step
A
→
B;
alternatively,
an
object
A
∈
Ob(C)
is
non-group-like
if
and
only
if
there
exists
a
co-angular
step
B
→
A.
Also,
if
A,
B
∈
Ob(C)
are
base-isomorphic
objects,
then
A
is
group-like
if
and
only
if
B
is.
Proof.
First,
we
consider
assertion
(i).
If
φ
∈
Arr(C)
is
a
pre-step,
and
the
projection
functor
C
→
D
is
full,
then
the
fact
that
it
is
a
linear
End-equivalence
follows
formally
from
the
definition
of
a
“pre-step”
[cf.
Definition
1.2,
(iii)];
the
fullness
assumption
on
C
→
D.
On
the
other
hand,
if
φ
∈
Arr(C)
is
a
linear
End-equivalence,
and
D
is
of
Aut-type,
then
it
follows
formally
that
φ
is
a
base-
isomorphism,
hence
a
pre-step,
as
desired.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
If
φ
∈
Arr(C)
is
a
co-angular
pre-step,
then
it
follows
formally
from
the
assumption
that
C
is
of
metrically
trivial
and
Aut-ample
type
that
φ
is
abstractly
equivalent
to
a
base-identity
pre-step
endomorphism
of
C.
On
the
other
hand,
if
φ
∈
Arr(C)
is
abstractly
equivalent
to
a
base-identity
pre-step
endomorphism
of
C
[hence
co-angular,
by
Definition
1.3,
(iii),
(b)],
then
it
follows
formally
that
φ
is
a
co-angular
linear
base-isomorphism,
hence
that
φ
is
a
co-angular
pre-step,
as
desired.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
observe
that
the
various
equivalences
of
assertion
(iii)
follow
formally
from
the
definitions
and
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d).
Proposition
1.9.
(Isotropic
Objects
and
Isometries)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobenioid.
THE
GEOMETRY
OF
FROBENIOIDS
I
31
Write
C
imtr-pre
⊆
C
for
the
subcategory
determined
by
the
isometric
pre-steps
and
imtr-pre
def
=
(C
imtr-pre
)
A
C
A
for
A
∈
Ob(C).
Then:
(i)
Any
base-isomorphism
φ
:
A
→
B
of
C
admits
a
factorization
φ
=
α
◦
β
where
α
is
an
isometric
pre-step,
and
β
is
a
co-angular
base-isomorphism;
this
factorization
is
unique,
up
to
replacing
the
pair
(α,
β)
by
a
pair
of
the
form
(α
◦
γ,
γ
−1
◦
β),
where
γ
is
an
isomorphism
of
C.
Here,
φ
is
isometric
if
and
only
if
β
is
a
morphism
of
Frobenius
type;
φ
is
co-angular
if
and
only
if
α
is
an
isomorphism;
φ
is
a
pull-back
morphism
if
and
only
if
φ
is
an
isomorphism.
(ii)
Any
base-isomorphism
φ
:
A
→
B
of
C
induces
a
functor
[well-defined
up
to
isomorphism]
imtr-pre
imtr-pre
φ
∗
:
C
A
→
C
B
that
maps
an
isometric
pre-step
C
→
A
to
the
isometric
pre-step
D
→
B
appearing
in
the
factorization
C
→
D
→
B
of
(i)
applied
to
the
composite
of
the
given
pre-step
C
→
A
with
φ
:
A
→
B.
Moreover,
if
φ
is
a
co-angular
pre-step,
then
φ
∗
is
an
equivalence
of
categories.
If
u
∈
O
×
(A),
then
we
shall
denote
by
u
imtr-pre
the
imtr-pre
induced
by
u
and
isomorphism
class
of
the
self-equivalence
of
the
category
C
A
by
O
×
(A)
imtr-pre
⊆
O
×
(A)
the
subgroup
of
v
∈
O
×
(A)
for
which
v
imtr-pre
is
the
identity.
(iii)
Any
pull-back
morphism
φ
:
A
→
B
of
C
induces
a
functor
[well-
defined
up
to
isomorphism]
imtr-pre
imtr-pre
φ
∗
:
C
B
→
C
A
that
maps
an
isometric
pre-step
δ
:
D
→
B
to
the
unique
[up
to
isomorphism]
isometric
pre-step
γ
:
C
→
A
that
fits
into
a
commutative
diagram
γ
C
−→
⏐
⏐
ψ
D
δ
−→
A
⏐
⏐
φ
B
where
ψ
is
the
pull-back
morphism
that
arises
by
applying
the
equivalence
of
cat-
egories
of
Definition
1.3,
(i),
(c),
to
the
arrow
Base(δ)
−1
◦
Base(φ),
and
γ
is
the
morphism
that
arises
from
the
isomorphism
of
functors
appearing
in
the
definition
of
a
“pull-back
morphism”
[cf.
Definition
1.2,
(ii)].
(iv)
Let
φ
:
A
→
B
be
a
co-angular
linear
morphism
[e.g.,
a
pull-back
morphism
—
cf.
Proposition
1.4,
(ii)].
Then
A
is
isotropic
if
and
only
if
B
is.
32
SHINICHI
MOCHIZUKI
(v)
C
istr
[equipped
with
the
restriction
to
C
of
the
given
functor
C
→
F
Φ
]
is
a
Frobenioid.
Moreover,
the
functor
C
→
C
istr
that
assigns
to
an
object
A
∈
Ob(C)
with
isotropic
hull
A
→
A
istr
the
object
A
istr
and
to
a
morphism
of
objects
A
→
B
with
isotropic
hulls
A
→
A
istr
,
B
→
B
istr
the
induced
[i.e.,
by
the
definition
of
an
“isotropic
hull”!]
morphism
A
istr
→
B
istr
forms
a
left
adjoint
to
the
inclusion
functor
C
istr
→
C,
through
which
the
functor
C
→
F
Φ
factors.
We
shall
refer
to
this
functor
as
the
isotropi-
fication
functor.
The
restriction
of
the
isotropification
functor
to
C
istr
is
iso-
morphic
to
the
identity
functor.
Finally,
the
isotropification
functor
preserves
morphisms
of
Frobenius
type,
Frobenius
degrees,
pre-steps,
pull-back
morphisms,
base-isomorphisms,
base-FSM-morphisms,
base-identity
en-
domorphisms,
Div-identity
endomorphisms,
isometries,
co-angular
mor-
phisms,
and
LB-invertible
morphisms;
moreover,
all
of
these
properties
are
compatible
with
the
inclusion
functor
C
istr
→
C
[in
the
sense
that
an
arrow
of
C
istr
satisfies
one
of
these
properties
with
respect
to
C
istr
if
and
only
if
it
does
with
respect
to
C].
(vi)
A
morphism
of
C
is
an
isotropic
hull
if
and
only
if
its
codomain
is
isotropic,
and,
moreover,
it
is
minimal-coadjoint
to
the
morphisms
with
isotropic
domain.
(vii)
A
morphism
A
→
B
of
C
is
an
isometric
pre-step
if
and
only
if
the
com-
posite
of
this
morphism
A
→
B
with
an
isotropic
hull
B
→
C
yields
an
isotropic
hull
A
→
C.
Proof.
Since
pull-backs
which
are
base-isomorphisms
are
easily
verified
to
be
isomorphisms
[cf.
Remark
1.2.1],
assertion
(i)
follows
immediately
from
the
(es-
sentially)
unique
factorization
of
Definition
1.3,
(iv),
(a);
the
(essentially)
unique
factorization
of
pre-steps
of
Definition
1.3,
(v),
(b);
the
fact
that
co-angular
mor-
phisms
are
closed
under
composition
[cf.
Proposition
1.7,
(i)];
the
definition
of
“co-angular”
[cf.
Definition
1.2,
(iii)];
the
fact
that
C
is
totally
epimorphic;
the
essential
uniqueness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Definition
1.3,
(ii)];
and
Remark
1.1.1.
Next,
we
consider
assertion
(ii).
The
existence
of
the
functor
φ
∗
follows
for-
mally
from
the
existence
of
the
(essentially)
unique
factorization
of
assertion
(i).
Now
suppose
that
φ
is
a
co-angular
pre-step.
Then
for
any
isometric
pre-step
β
:
D
→
B,
there
exists
a
co-angular
pre-step
ψ
:
C
→
D
such
that
(Φ(β
◦
ψ))
−1
(Div(ψ)))
=
(Φ(φ))
−1
(Div(φ))
[cf.
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d)].
Thus,
by
applying
the
factorization
of
Definition
1.3,
(v),
(c),
it
follows
that
we
may
write
β
◦
ψ
=
φ
◦
α
,
where
α
:
D
→
A
is
an
isometric
pre-step,
and
φ
:
A
→
B
is
a
THE
GEOMETRY
OF
FROBENIOIDS
I
33
co-angular
pre-step.
On
the
other
hand,
since
Div(β
◦
ψ)
=
Div(φ
◦
α
),
and
β,
α
are
isometric,
it
follows
that
(Φ(φ))
−1
(Div(φ))
=
(Φ(φ
))
−1
(Div(φ
))
—
hence
[by
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d)]
that
∼
there
exists
an
isomorphism
γ
:
A
→
A
such
that
φ
◦
γ
=
φ
.
Thus,
if
we
take
def
α
=
γ
◦α
,
then
β
◦ψ
=
φ◦α
—
that
is
to
say,
φ
∗
is
essentially
surjective.
Moreover,
[by
possibly
replacing
φ
by
ψ]
this
argument
[i.e.,
the
construction,
given
β,
φ,
of
α,
ψ
such
that
β
◦
ψ
=
φ
◦
α]
also
implies
that
φ
∗
is
full.
Finally,
since
every
pre-step
is
a
monomorphism
[cf.
Definition
1.3,
(v),
(a)],
it
follows
immediately
that
φ
∗
is
faithful.
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
formally
from
the
definitions,
together
with
the
fact
that
pull-back
morphisms
are
linear
isometries
[cf.
Proposition
1.4,
(ii)],
which
implies
[cf.
Remark
1.1.1]
that
γ
is
an
isometric
pre-step.
Next,
we
consider
assertion
(iv).
Let
φ
:
A
→
B
be
a
co-angular
linear
mor-
phism.
If
A
is
isotropic,
then
so
is
B,
by
Definition
1.3,
(vii),
(b).
Now
suppose
that
B
is
isotropic.
Thus,
by
the
definition
of
an
isotropic
hull,
it
follows
from
the
existence
of
isotropic
hulls
[cf.
Definition
1.3,
(vii),
(a)]
that
there
exists
a
factor-
ization
φ
=
β
◦
α,
where
α
:
A
→
A
is
an
isotropic
hull
[hence
an
isometric
pre-step
—
cf.
Definition
1.2,
(iv)],
and
β
:
A
→
B
is
linear
[cf.
Remark
1.1.1].
Thus,
by
the
definition
of
“co-angular”
[cf.
Definition
1.2,
(iii)],
we
conclude
that
α
is
an
isomorphism,
as
desired.
This
completes
the
proof
of
assertion
(iv).
Next,
we
consider
assertion
(v).
By
applying
the
definition
of
an
isotropic
hull
[cf.
Definition
1.2,
(iv)],
it
follows
immediately
[from
the
fact
that
C
is
connected
and
totally
epimorphic]
that
C
istr
is
connected
and
totally
epimorphic.
Thus,
C
istr
is
a
pre-Frobenioid.
It
is
immediate
from
the
definition
of
an
isotropic
hull
that
the
isotropification
functor
is
left
adjoint
to
the
inclusion
functor
C
istr
→
C;
that
the
functor
C
→
F
Φ
factors
through
the
isotropification
functor
[cf.
Remark
1.1.1];
that
the
restriction
of
the
isotropification
functor
to
C
istr
is
isomorphic
to
the
identity
functor;
and
[cf.
Remark
1.1.1]
that
the
isotropification
functor
preserves
Frobe-
nius
degrees,
pre-steps,
base-isomorphisms,
base-FSM-morphisms,
base-identity
en-
domorphisms,
Div-identity
endomorphisms,
isometries,
and
co-angular
morphisms
[cf.
Proposition
1.4,
(i)],
hence
also
LB-invertible
morphisms
and
morphisms
of
Frobenius
type
in
a
fashion
that
is
compatible
[cf.
the
statement
of
assertion
(v)]
with
the
inclusion
C
istr
→
C.
Since
pull-back
morphisms
are
co-angular
linear
isometries
[cf.
Proposition
1.4,
(ii)],
it
follows
immediately
[in
light
of
what
we
have
shown
so
far]
from
Proposition
1.4,
(ii),
that
the
isotropification
functor
maps
pull-back
morphisms
to
morphisms
which
are
pull-back
morphisms
relative
to
C,
hence
a
fortiori,
pull-back
morphisms
relative
to
C
istr
.
Finally,
in
light
of
Proposi-
tion
1.4,
(i);
assertion
(iv)
[cf.
also
Definition
1.3,
(vii),
(b)],
it
follows
immediately
[from
the
fact
that
C
is
a
Frobenioid!]
that
the
pre-Frobenioid
C
istr
satisfies
the
various
conditions
of
Definition
1.3,
hence
that
C
istr
is
a
Frobenioid,
as
desired.
This
completes
the
proof
of
assertion
(v).
Finally,
we
observe
that
the
necessity
and
sufficiency
of
the
condition
of
asser-
tion
(vi)
follow
immediately
from
the
definition
of
an
isotropic
hull
[cf.
Definition
34
SHINICHI
MOCHIZUKI
1.2,
(iv)],
the
existence
of
isotropic
hulls
[cf.
Definition
1.3,
(vii),
(a)]
and
the
total
epimorphicity
of
C;
the
necessity
and
sufficiency
of
the
condition
of
assertion
(vii)
follow
immediately
from
the
existence
of
isotropic
hulls
[cf.
Definition
1.3,
(vii),
(a)],
the
fact
that
isometric
pre-steps
between
isotropic
objects
are
isomorphisms
[cf.
Definition
1.3,
(vii),
(b);
Proposition
1.4,
(i),
(iii)],
and
the
following
observa-
tion
[which
follows
immediately
from
Proposition
1.7,
(i),
(v)]:
Given
morphisms
α,
β,
γ
of
C
such
that
γ
=
α
◦
β,
if
any
two
of
the
three
morphisms
α,
β,
γ
is
an
isometric
pre-step,
then
the
same
is
true
of
the
remaining
morphism.
Proposition
1.10.
(Morphisms
of
Frobenius
Type)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobenioid.
Then:
(i)
Let
φ
:
A
→
B
be
an
arbitrary
morphism
of
C.
Suppose
that
α
:
A
→
A
,
β
:
B
→
B
are
morphisms
of
Frobenius
type,
of
Frobenius
degree
d
∈
N
≥1
.
Then
there
exists
a
unique
morphism
φ
:
A
→
B
such
that
the
following
diagram
commutes:
φ
A
−→
B
⏐
⏐
⏐
β
⏐
α
A
φ
−→
B
In
this
situation,
deg
Fr
(φ)
=
deg
Fr
(φ
);
Div(φ
)
=
d
·
α
∗
(Div(φ))
[where
we
write
∼
α
∗
:
Φ(A)
→
Φ(A
)
for
the
bijection
induced
by
applying
the
functor
Φ
to
the
base-
isomorphism
α].
Finally,
if
φ
is
a
morphism
of
Frobenius
type
(respectively,
pre-step;
pull-back
morphism;
co-angular
morphism;
base-isomorphism;
isometry;
LB-invertible
morphism),
then
the
same
is
true
of
φ
.
(ii)
Any
composite
morphism
β
◦
α
of
C,
where
α
is
a
pre-step,
and
β
is
of
Frobenius
type,
may
be
written
as
a
composite
α
◦
β
=
β
◦
α
where
α
is
a
pre-step,
and
β
is
of
Frobenius
type
such
that:
deg
Fr
(β)
=
deg
Fr
(β
);
Div(α
)
=
deg
Fr
(β)
·
β
∗
(Div(α))
[where
we
write
β
∗
for
the
bijection
induced
by
applying
the
functor
Φ
to
the
base-
isomorphism
β
].
(iii)
Suppose
that
C
is
of
perfect
type.
Then
the
monoids
in
the
image
of
Φ
are
perfect.
If,
moreover,
C
is
of
isotropic
and
Frobenius-normalized
type,
then
the
monoids
O
(A)
and
O
×
(A)
are
perfect.
(iv)
A
morphism
of
Frobenius
type
with
isotropic
domain
is
a
prime-Frobe-
nius
morphism
if
and
only
if
it
is
irreducible
[cf.
§0].
In
particular,
if
A
∈
Ob(C)
is
isotropic,
then
there
exist
infinitely
many
isomorphism
classes
of
objects
of
A
C
that
arise
from
irreducible
arrows
with
domain
A.
THE
GEOMETRY
OF
FROBENIOIDS
I
35
(v)
A
morphism
of
C
is
a
morphism
of
Frobenius
type
if
and
only
if
it
is
a
composite
of
prime-Frobenius
morphisms.
(vi)
The
Frobenioid
C
istr
is
of
sub-quasi-Frobenius-trivial
type.
Moreover,
every
group-like
object
A
∈
Ob(C
istr
)
is
Frobenius-trivial.
Proof.
First,
we
consider
assertion
(i).
Observe
that
uniqueness
follows
from
the
fact
that
C
is
totally
epimorphic.
Now
it
suffices
to
prove
the
existence
of
φ
as
desired,
first
in
the
case
where
φ
is
a
morphism
of
Frobenius
type,
then
in
the
case
where
φ
is
a
pre-step,
and
finally
in
the
case
where
φ
is
a
pull-back
morphism
[cf.
the
factorization
of
Definition
1.3,
(iv),
(a)].
In
the
first
case,
since
morphisms
of
Frobenius
type
are
closed
under
composition,
with
multiplying
Frobenius
degrees
[cf.
Proposition
1.7,
(i);
Remark
1.1.1],
the
existence
of
a
morphism
of
Frobenius
type
φ
as
desired
follows
immediately
from
the
existence
and
(essential)
uniqueness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Definition
1.3,
(ii)].
In
the
case
where
φ
is
a
pre-step,
the
existence
of
a
pre-step
φ
[which,
moreover,
is
co-angular
if
φ
is]
as
desired
follows
immediately
from
the
factorization
of
Definition
1.3,
(iv),
(a)
[cf.
also
Proposition
1.4,
(iv)],
together
with
the
(essential)
unique-
ness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Definition
1.3,
(ii)],
and
the
fact
that
co-angular
morphisms
are
closed
under
composition
[cf.
Proposition
1.7,
(i)].
In
a
similar
vein,
since
pull-back
morphisms
are
LB-invertible
[cf.
Proposition
1.4,
(ii)],
and
LB-invertible
morphisms
are
closed
under
compo-
sition
[cf.
Proposition
1.7,
(i)],
the
existence
of
a
pull-back
morphism
φ
in
the
case
where
φ
is
a
pull-back
morphism
follows
immediately
from
the
factorization
of
Proposition
1.4,
(v),
together
with
the
(essential)
uniqueness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Definition
1.3,
(ii)].
The
portion
of
assertion
(i)
concerning
“deg
Fr
(−)”,
“Div(−)”
then
follows
immediately
from
Remark
1.1.1.
Finally,
in
light
of
what
we
have
done
so
far,
the
fact
that
“if
φ
is
a(n)
co-angular
morphism
(respectively,
base-isomorphism;
isometry;
LB-invertible
morphism),
then
the
same
is
true
of
φ
”
follows
immediately
from
the
definitions;
Remark
1.1.1;
the
factorization
of
co-angular
morphisms
given
in
Proposition
1.4,
(iv);
and
the
fact
that
co-angular
morphisms
are
closed
under
composition
[cf.
Proposition
1.7,
(i)].
This
completes
the
proof
of
assertion
(i).
Now
[in
light
of
the
existence
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
—
cf.
Definition
1.3,
(ii)]
assertion
(ii)
follows
formally
from
assertion
(i).
Next,
we
consider
assertion
(iii).
In
light
of
the
existence
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Definition
1.3,
(ii)]
and
the
equiva-
lences
of
categories
of
Definition
1.3,
(iii),
(d),
the
fact
that
Φ(A)
is
perfect
follows
immediately
[cf.
Remark
1.1.1]
from
the
fact
that
A
is
perfect
[cf.
Definition
1.2,
(iv)].
Now
suppose
further
that
C
is
of
isotropic
[so
all
morphisms
of
C
are
co-angular
—
cf.
Proposition
1.4,
(i)]
and
Frobenius-normalized
type.
Then
by
the
existence
of
Frobenius-trivial
objects
[cf.
Definition
1.3,
(i),
(a),
(b);
the
isomorphism
of
Def-
inition
1.3,
(iii),
(c)],
we
may
assume
that
A
is
Frobenius-trivial.
Now
the
fact
that
the
monoids
O
(A)
and
O
×
(A)
are
perfect
follows
immediately
from
the
fact
that
A
is
perfect
[cf.
Definition
1.2,
(iv),
applied
to
the
base-identity
endomorphisms
of
Frobenius
type
of
the
Frobenius-trivial
object
A]
and
Frobenius-normalized.
This
completes
the
proof
of
assertion
(iii).
36
SHINICHI
MOCHIZUKI
Next,
we
observe
that
assertion
(iv)
follows
immediately
from
Proposition
1.7,
(v),
and
the
well-known
structure
of
the
multiplicative
monoid
N
≥1
[cf.
also
Def-
inition
1.3,
(ii)],
and
that
assertion
(v)
follows
immediately
from
Proposition
1.7,
(i);
Definition
1.3,
(ii).
Finally,
we
consider
assertion
(vi).
Let
A
∈
Ob(C
istr
).
Then
by
Definition
1.3,
(i),
(a),
(b)
[applied
to
the
Frobenioid
C
istr
—
cf.
Proposition
1.9,
(v)],
there
exist
co-angular
[cf.
Proposition
1.4,
(i)]
pre-steps
α
:
B
→
A,
γ
:
B
→
C,
where
C
is
Frobenius-trivial.
Thus,
for
d
∈
N
≥1
,
there
exists
a
base-identity
endomorphism
of
Frobenius
type
φ
C
∈
End
C
(C)
such
that
deg
Fr
(φ
C
)
=
d;
by
assertion
(ii)
[cf.
also
Proposition
1.4,
(i)],
we
may
write
φ
C
◦
γ
=
γ
◦
ψ,
where
ψ
:
B
→
B
is
a
morphism
of
Frobenius
type,
and
γ
:
B
→
C
is
a
co-angular
pre-step.
Moreover,
the
portion
of
assertion
(ii)
concerning
the
relationship
between
Div(γ),
Div(γ
)
implies,
in
light
of
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d),
that
γ
factors
through
γ,
i.e.,
there
exists
a
co-angular
pre-step
β
:
B
→
B
such
that
def
γ
◦
β
=
γ
.
Thus,
if
we
set
φ
B
=
β
◦
ψ
∈
End
C
(B),
then
γ
◦
φ
B
=
φ
C
◦
γ.
Moreover,
since
φ
C
is
a
base-identity
endomorphism
of
Frobenius
degree
d,
and
γ
is
a
pre-
step,
it
follows
[cf.
Remark
1.1.1]
that
φ
B
is
also
a
base-identity
endomorphism
of
Frobenius
degree
d.
Thus,
we
conclude
that
B
is
quasi-Frobenius-trivial,
hence
that
A
is
sub-quasi-Frobenius-trivial,
as
desired.
If,
moreover,
A
is
group-like,
then
[since
C
istr
is
a
Frobenioid
—
cf.
Proposition
1.9,
(v)]
it
follows
from
Definition
1.3,
(i),
(a),
(b),
that
there
exist
[co-angular
—
cf.
Proposition
1.4,
(i)]
pre-steps
A
→
A,
A
→
A
,
where
A
is
Frobenius-trivial.
But
by
Proposition
1.4,
(iii),
these
pre-
steps
are
isomorphisms,
so
A
is
Frobenius-trivial,
as
desired.
This
completes
the
proof
of
assertion
(vi).
Proposition
1.11.
(Pull-back
and
Linear
Morphisms)
Let
Φ
be
a
divi-
sorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobe-
nioid.
Then:
(i)
Suppose
further
that
C
is
of
Aut-ample
and
base-trivial
type.
Then
the
natural
projection
functor
C
pl-bk
→
D
is
full.
(ii)
Suppose
further
that
C
is
of
unit-trivial
type.
Then
the
natural
projection
functor
C
pl-bk
→
D
is
faithful.
(iii)
Let
φ
:
B
→
A
be
a
pull-back
morphism
that
projects
to
a
morphism
def
φ
D
=
Base(φ)
:
B
D
→
A
D
of
D
.
Then
given
any
α
∈
End
C
(A),
β
D
∈
End
D
(B
D
)
such
that
Base(α)
◦
φ
D
=
φ
D
◦
β
D
,
there
exists
a
unique
β
∈
End
C
(B)
such
that
Base(β)
=
β
D
,
α
◦
φ
=
β
◦
φ.
(iv)
Every
co-angular
linear
morphism
φ
:
B
→
A
determines
an
injection
of
monoids
O
(A)
→
O
(B)
which
is
uniquely
determined
by
the
condition
that
O
(A)
implies
α
◦
φ
=
φ
◦
β.
α
→
β
∈
O
(B)
THE
GEOMETRY
OF
FROBENIOIDS
I
37
(v)
The
equivalences
of
categories
of
Definition
1.3,
(iii),
(d),
are
“functorial”
in
the
following
sense:
If
φ
:
A
→
B
is
an
arbitrary
morphism
of
C
lin
,
α
:
C
→
A
and
β
:
D
→
B
(respectively,
α
:
A
→
C
and
β
:
B
→
D)
are
co-angular
pre-steps
such
that
(α
∗
)
−1
(Div(α))
=
φ
∗
{(β
∗
)
−1
(Div(β))}
(respectively,
Div(α)
=
φ
∗
(Div(β))),
then
there
exists
a
unique
morphism
ψ
:
C
→
D
in
C
lin
such
that
β
◦
ψ
=
φ
◦
α
(respectively,
ψ
◦
α
=
β
◦
φ).
Moreover,
φ
is
a
pull-back
morphism
if
and
only
if
ψ
is.
(vi)
A
pull-back
morphism
φ
∈
Arr(C)
is
an
FSM-morphism
(respectively,
fiberwise-surjective
morphism;
monomorphism;
irreducible
morphism)
if
and
only
if
Base(φ)
∈
Arr(D)
is.
(vii)
Let
φ
:
A
→
B
be
a
co-angular
pre-step;
:
C
→
B
a
morphism.
Then
there
exists
a
co-angular
pre-step
γ
:
D
→
C
and
a
morphism
α
:
D
→
A
such
that
◦
γ
=
φ
◦
α.
In
particular,
every
co-angular
pre-step
of
C
is
an
FSM-morphism.
def
Proof.
First,
we
consider
assertion
(i).
Let
A,
B
∈
Ob(C);
A
D
=
Base(A);
def
B
=
Base(B);
φ
D
:
A
D
→
B
D
a
morphism
in
D.
By
the
equivalence
of
categories
of
Definition
1.3,
(i),
(c),
it
follows
that
there
exists
a
pull-back
morphism
ψ
:
C
→
B
def
of
C
such
that
ψ
D
=
Base(ψ)
:
C
D
→
B
D
of
D
defines
an
object
of
D
B
D
that
is
isomorphic
to
the
object
defined
by
φ
D
.
In
particular,
C
D
is
isomorphic
to
A
D
.
Since
C
is
of
base-trivial
type,
it
thus
follows
that
A,
C
are
isomorphic,
so
we
may
assume
that
A
=
C.
Thus,
ψ
projects
to
a
morphism
ψ
D
:
A
D
→
B
D
of
D
such
that
φ
D
=
ψ
D
◦
δ,
for
some
δ
∈
Aut
D
(A
D
).
Since
C
is
of
Aut-ample
type,
it
thus
follows
that
δ
lifts
to
a
γ
∈
Aut
C
(A).
Thus,
taking
ψ
◦
γ
:
A
→
B
yields
a
morphism
of
C
that
projects
to
φ
D
.
This
completes
the
proof
of
assertion
(i).
def
def
Next,
we
consider
assertion
(ii).
Let
A,
B
∈
Ob(C);
A
D
=
Base(A);
B
D
=
Base(B);
φ,
ψ
:
A
→
B
pull-back
morphisms
of
C
that
project
to
the
same
morphism
A
D
→
B
D
of
D.
By
the
definition
of
a
“pull-back
morphism”
[cf.
Definition
1.2,
(ii)],
it
thus
follows
formally
that
there
exist
base-identity
endomorphisms
α,
β
∈
End
C
(A)
such
that
ψ
=
φ
◦
α,
φ
=
ψ
◦
β.
In
particular,
we
obtain
that
ψ
=
ψ
◦
β
◦
α,
φ
=
φ
◦
α
◦
β,
hence
[again
by
Definition
1.2,
(ii)]
that
α
◦
β,
β
◦
α
are
both
equal
to
the
identity
endomorphism
of
A,
i.e.,
that
α,
β
∈
Aut
C
(A).
But
this
implies
that
α,
β
∈
O
×
(A)
=
{1},
so
φ
=
ψ,
as
desired.
This
completes
the
proof
of
assertion
(ii).
Next,
we
consider
assertion
(iii).
The
existence
and
uniqueness
of
β
as
asserted
follows
immediately
from
the
isomorphism
of
functors
appearing
in
the
definition
of
a
“pull-back
morphism”
[cf.
Definition
1.2,
(ii)].
This
completes
the
proof
of
assertion
(iii).
Now
since
a
co-angular
linear
morphism
factors
as
the
composite
of
a
pull-back
morphism
with
a
co-angular
pre-step
[cf.
Propositions
1.4,
(iv);
1.7,
(iii)],
the
existence
of
the
map
“→”
of
assertion
(iv)
follows
immediately
[cf.
Proposition
1.7,
(iii)]
from
assertion
(iii)
and
Definition
1.3,
(iii),
(c);
the
asserted
injectivity
of
this
map
follows
from
the
total
epimorphicity
of
C;
the
fact
that
this
map
is
uniquely
determined
by
the
condition
given
in
assertion
(iii)
follows
from
the
fact
that
pre-steps
are
monomorphisms
[cf.
Definition
1.3,
(v),
(a)],
and
the
definition
of
a
“pull-back
morphism”
in
Definition
1.2,
(ii).
38
SHINICHI
MOCHIZUKI
Next,
we
consider
assertion
(v).
First,
we
observe
that
the
uniqueness
of
ψ
follows
from
the
fact
that
β
is
a
monomorphism
[cf.
Definition
1.3,
(v),
(a)]
in
the
non-resp’d
case
and
from
the
total
epimorphicity
of
C
applied
to
α
in
the
resp’d
case.
When
φ
is
a
pull-back
morphism
[hence
co-angular
and
linear
—
cf.
Proposition
1.4,
(ii)],
the
existence
of
a
pull-back
morphism
ψ
as
desired
follows
immediately
by
applying
the
equivalence
of
categories
induced
by
the
projection
functor
in
Def-
inition
1.3,
(i),
(c);
the
definition
of
a
“pull-back
morphism”
in
Definition
1.2,
(ii);
Proposition
1.7,
(i),
(v)
[applied
to
co-angular
linear
morphisms];
and
the
equiva-
lences
of
categories
of
Definition
1.3,
(iii),
(d).
When
φ
is
an
isometric
pre-step,
the
existence
of
an
isometric
pre-step
ψ
as
desired
follows
immediately
from
the
equiv-
alence
of
categories
of
Proposition
1.9,
(ii)
[in
the
“case
of
a
co-angular
pre-step”].
When
φ
is
a
co-angular
pre-step,
the
existence
of
a
co-angular
pre-step
ψ
as
desired
follows
formally
from
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d).
In
light
of
the
factorizations
of
Definition
1.3,
(v),
(b),
(c);
Proposition
1.7,
(iii),
this
completes
the
proof
of
assertion
(v).
Next,
we
observe
that
assertion
(vi)
follows
formally
from
the
isomorphism
of
functors
appearing
in
the
definition
of
a
“pull-back
morphism”
[cf.
Definition
1.2,
(ii)],
together
with
the
equivalence
of
categories
induced
by
the
projection
functor
in
Definition
1.3,
(i),
(c)
[cf.
also
Proposition
1.7,
(v),
for
pull-back
morphisms].
Finally,
we
consider
assertion
(vii).
By
applying
the
factorizations
of
Definition
1.3,
(iv),
(a);
Definition
1.3,
(v),
(b),
it
follows
immediately
that
we
may
assume
without
loss
of
generality
[from
the
point
of
view
of
showing
the
existence
of
γ,
α
with
the
desired
properties]
that
is
a
pull-back
morphism,
an
isometric
pre-
step,
a
co-angular
pre-step,
or
a
morphism
of
Frobenius
type.
If
is
a
pull-back
morphism,
then
it
follows
immediately
[by
“pulling
back
the
zero
divisor
of
φ
via
”
—
cf.
assertion
(v)]
that
there
exist
a
pull-back
morphism
α
:
D
→
A
and
a
co-angular
pre-step
γ
:
D
→
C
such
that
◦
γ
=
φ
◦
α.
Next,
observe
that
if
is
an
isometric
pre-step,
then
the
existence
of
γ,
α
with
the
desired
properties
follows
formally
from
the
equivalence
of
categories
of
Proposition
1.9,
(ii)
[induced
by
φ].
Next,
observe
that
if
is
a
co-angular
pre-step,
then
it
follows
immediately
from
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d),
that
there
exist
co-angular
pre-steps
α
:
D
→
A,
γ
:
D
→
C
such
that
◦
γ
=
φ
◦
α.
Finally,
we
consider
the
case
where
is
a
morphism
of
Frobenius
type.
By
applying
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d),
it
follows
that
we
may
assume
[by
replacing
φ
by
the
composite
of
φ
with
an
appropriate
pre-step
A
→
A]
that
Div(φ)
=
deg
Fr
(
)
·
x,
for
some
x
∈
Φ(A).
Thus,
[by
applying
again
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d)],
it
follows
that
there
exist
a
morphism
of
Frobenius
type
α
:
D
→
A
and
a
co-angular
pre-step
γ
:
D
→
C
such
that
◦
γ
=
φ
◦
α
[cf.
also
Proposition
1.10,
(i)].
This
completes
the
proof
of
the
existence
of
γ,
α
with
the
desired
properties.
It
thus
follows
formally
that
every
co-angular
pre-step
of
C
is
fiberwise
surjective.
On
the
other
hand,
by
Definition
1.3,
(v),
(a),
every
pre-step
is
a
monomorphism.
Thus,
we
conclude
that
every
co-
angular
pre-step
of
C
is
an
FSM-morphism.
This
completes
the
proof
of
assertion
(vii).
Remark
1.11.1.
Observe
that
in
the
situation
of
Proposition
1.11,
(iii),
if
α
is
THE
GEOMETRY
OF
FROBENIOIDS
I
39
a
morphism
of
Frobenius
type,
and
β
D
is
an
isomorphism,
then
β
is
a
morphism
of
Frobenius
type.
[Indeed,
then
β
is
co-angular
by
Definition
1.3,
(iii),
(b),
and
isometric
by
Remark
1.1.1.]
In
particular,
it
follows
[cf.
Remark
1.2.1]
that
[at
least
in
the
case
of
Frobenioids]
“Frobenius-trivial”
implies
“universally
Div-Frobenius-
trivial”.
Proposition
1.12.
(Endomorphisms)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobenioid;
A
∈
Ob(C);
def
A
D
=
Base(A)
∈
Ob(D).
Then:
(i)
We
have
natural
exact
sequences
of
monoids
1
→
O
×
(A)
→
Aut
C
(A)
→
Aut
D
(A
D
)
1
→
O
(A)
→
End
C
(A)
→
N
≥1
×
End
D
(A
D
)
—
where
the
second
arrow
in
each
sequence
is
the
natural
inclusion;
the
third
ar-
row
of
the
first
sequence
is
determined
by
the
natural
projection
functor
to
D;
the
third
arrow
of
the
second
sequence
is
determined
by
the
Frobenius
degree
and
the
natural
projection
functor
to
D.
If,
moreover,
A
is
Aut-ample
(respectively,
End-
ample;
quasi-Frobenius-trivial),
then
the
map
Aut
C
(A)
→
Aut
D
(A
D
)
(respec-
tively,
End
C
(A)
→
End
D
(A
D
);
End
C
(A)
→
N
≥1
)
is
surjective.
(ii)
An
endomorphism
of
A
is
a
sub-automorphism
[cf.
§0]
if
and
only
if
it
is
an
isometric
linear
endomorphism
that
projects
to
a
sub-automorphism
of
D.
(iii)
A
sub-automorphism
of
A
is
an
automorphism
if
and
only
if
it
is
a
base-
isomorphism.
(iv)
Suppose
that
A
is
Aut
sub
-ample.
Then
A
is
Aut-saturated
[cf.
§0]
if
and
only
if
A
D
is.
Proof.
Assertion
(i)
is
immediate
from
the
definitions.
The
necessity
of
the
condi-
tions
of
assertion
(ii),
(iii)
is
immediate
from
Remark
1.1.1.
To
prove
the
sufficiency
of
the
conditions
of
assertion
(ii),
(iii),
it
suffices,
in
light
of
the
equivalence
of
cate-
gories
[involving
pull-back
morphisms]
of
Definition
1.3,
(i),
(c)
[cf.
also
Proposition
1.11,
(iii)],
and
the
fact
that
endomorphisms
are
always
co-angular
[cf.
Definition
1.3,
(iii),
(b)],
to
observe
that
any
LB-invertible
linear
base-isomorphism
[i.e.,
LB-
invertible
pre-step]
is,
in
fact,
an
isomorphism
[cf.
Proposition
1.4,
(iii)].
Now
assertion
(iv)
follows
formally
from
assertions
(ii),
(iii)
and
the
definitions.
Proposition
1.13.
(Rigidity
and
Slimness)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobenioid;
A
∈
Ob(C);
def
A
D
=
Base(A)
∈
Ob(D).
Suppose
further
that
the
category
D
is
slim
[cf.
§0].
Then:
40
SHINICHI
MOCHIZUKI
(i)
The
composite
C
A
→
D
of
the
natural
functor
C
A
→
C
with
the
natural
projection
functor
C
→
D
is
rigid
[cf.
§0].
In
particular,
the
functor
C
→
D
is
rigid.
(ii)
The
composite
C
A
→
F
Φ
of
the
natural
functor
C
A
→
C
with
the
functor
C
→
F
Φ
is
rigid.
In
particular,
the
functor
C
→
F
Φ
is
rigid.
(iii)
Suppose,
moreover,
that
every
object
A
∈
Ob(C)
satisfies
[at
least]
one
of
the
following
two
conditions:
(a)
O
×
(A)
imtr-pre
=
{1}
[cf.
Proposition
1.9,
(ii)];
(b)
n∈N
≥1
{O
×
(A)}
n
=
{1},
and,
moreover,
there
exists
a
co-angular
pre-step
∼
B
→
A
[which,
by
Definition
1.3,
(iii),
(c),
induces
a
bijection
O
×
(B)
→
O
×
(A)]
such
that
B
is
quasi-Frobenius-trivial
and
Frobenius-normalized.
Then
the
category
C
is
slim.
Proof.
First,
we
consider
assertion
(i).
Any
automorphism
α
of
the
functor
C
A
→
pl-bk
→
C
A
→
D.
On
D
determines
an
automorphism
of
the
composite
functor
C
A
pl-bk
→
D
A
D
→
D,
the
other
hand,
this
composite
functor
factors
as
a
composite
C
A
pl-bk
where
the
first
functor
C
A
→
D
A
D
is
[by
Definition
1.3,
(i),
(c)]
an
equivalence
of
categories.
Thus,
we
conclude
that
α
determines
an
automorphism
of
the
natural
functor
D
A
D
→
D,
which
is
necessarily
trivial,
since
D
is
slim.
Since
A
is
arbitrary,
we
thus
conclude
that
both
C
A
→
D
and
C
→
D
are
rigid.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
Let
α
be
an
automorphism
of
the
functor
C
A
→
F
Φ
.
By
assertion
(i),
it
follows
that
the
automorphisms
of
objects
of
F
Φ
[which,
by
Proposition
1.5,
(i),
is
itself
a
Frobenioid]
induced
by
α
are
base-identity
automorphisms.
Since
Φ
is
divisorial,
hence,
in
particular,
sharp
[cf.
Definition
1.1,
(i),
(ii)],
it
thus
follows
that
all
of
these
automorphisms
are
trivial,
hence
that
α
is
trivial.
Since
A
is
arbitrary,
we
thus
conclude
that
both
C
A
→
F
Φ
and
C
→
F
Φ
are
rigid.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
consider
assertion
(iii).
Let
α
be
an
automorphism
of
the
natural
functor
C
A
→
C.
By
assertion
(i),
it
follows
that
the
automorphisms
of
objects
of
C
induced
by
α
are
base-identity
automorphisms,
i.e.,
belong
to
“O
×
(−)”.
Moreover,
the
functoriality
of
the
automorphisms
induced
by
α
with
respect
to
isometric
pre-
steps
implies
that
these
automorphisms
belong
to
“O
×
(−)
imtr-pre
”.
Similarly,
the
functoriality
of
the
automorphisms
induced
by
α
with
respect
to
base-identity
en-
domorphisms
implies
that,
at
least
in
the
case
of
quasi-Frobenius-trivial,
Frobenius-
normalized
objects
—
hence
also
[cf.
Definition
1.3,
(iii),
(c)]
objects
as
in
(b)
of
the
statement
of
assertion
(iii)
—
these
automorphisms
belong
to
“
n∈N
≥1
{O
×
(−)}
n
”.
Thus,
we
conclude
that
under
either
of
the
assumptions
(a),
(b)
in
the
statement
of
assertion
(iii),
the
automorphisms
induced
by
α
are
trivial.
This
completes
the
proof
of
assertion
(iii).
Remark
1.13.1.
Note
that
if
the
hypothesis
of
Proposition
1.13,
(iii),
fails
to
hold,
then
it
is
not
necessarily
the
case
that
C
is
slim.
Indeed,
if
M
is
a
perfect
THE
GEOMETRY
OF
FROBENIOIDS
I
41
pre-divisorial
monoid,
and
C
is
a
one-object
category
whose
unique
object
has
endo-
morphism
monoid
equal
to
the
elementary
Frobenioid
F
M
[so
C
equipped
with
the
functor
of
one-object
categories
determined
by
the
natural
morphism
of
monoids
F
M
→
F
M
char
is
a
Frobenioid,
by
Proposition
1.5,
(i)],
then
any
collection
of
ele-
ments
{α
n
}
n∈N
≥1
of
M
±
such
that
α
nm
=
m
·
α
n
for
all
n,
m
∈
N
≥1
determines
an
automorphism
of
the
natural
functor
C
A
→
C
[which
is
nontrivial
as
soon
as
any
of
the
α
n
is
nonzero]
by
assigning
to
an
arrow
φ
:
B
→
A
of
C
the
automorphism
α
deg
Fr
(φ)
∈
Aut
C
(B).
One
key
result
for
analyzing
the
category-theoretic
structure
of
Frobenioids
[cf.
§3]
is
the
following:
Proposition
1.14.
(Irreducible
Morphisms)
Let
Φ
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D;
C
→
F
Φ
a
Frobenioid
of
isotropic
type;
φ
∈
Arr(C).
Suppose
further
that
D
is
of
FSMFF-type
[cf.
§0].
Then:
(i)
φ
is
irreducible
if
and
only
if
φ
is
one
of
the
following:
(a)
a
prime-
Frobenius
morphism;
(b)
a
step
such
that
Div(φ)
is
irreducible;
(c)
a
pull-back
morphism
such
that
Base(φ)
is
an
irreducible
morphism
of
D.
(ii)
φ
is
a
pre-step
if
and
only
if
it
is
an
FSM-morphism
that
is
mid-
adjoint
[cf.
§0]
to
the
irreducible
morphisms
which
are
not
pre-steps.
(iii)
Suppose
that
φ
is
irreducible.
Then
φ
is
a
non-pre-step
if
and
only
if
the
following
condition
holds:
There
exists
an
N
∈
N
≥1
such
that
for
every
equality
of
composites
in
C
α
n
◦
α
n−1
◦
.
.
.
◦
α
2
◦
α
1
=
ψ
◦
φ
—
where
α
1
,
.
.
.
,
α
n
,
ψ
are
FSMI-morphisms
[cf.
§0]
—
it
holds
that
n
≤
N
.
(iv)
Let
α
◦
β
=
β
◦
α
be
an
equality
of
composites
of
C,
where
deg
Fr
(β)
=
deg
Fr
(β
),
and
α,
α
are
irreducible.
Then
α
is
a
prime-Frobenius
morphism
if
and
only
if
α
is;
moreover,
deg
Fr
(α)
=
deg
Fr
(α
).
(v)
Suppose
further
that
Φ
is
non-dilating,
and
that
φ
is
a
non-pre-step
irreducible
endomorphism
of
a
non-group-like
object
A
∈
Ob(C).
Then
φ
is
a
Div-identity
prime-Frobenius
endomorphism
if
and
only
if
the
following
condition
holds:
For
every
step
α
:
A
→
B,
there
exists
a
non-pre-step
irreducible
morphism
ψ
:
B
→
B
and
a
step
β
:
B
→
B
such
that
ψ
◦
α
=
β
◦
α
◦
φ.
Proof.
First,
we
consider
assertion
(i).
The
sufficiency
of
the
condition
of
assertion
(i)
follows
for
morphisms
as
in
(a)
(respectively,
(b);
(c))
from
Proposition
1.10,
(iv)
(respectively,
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d)
[cf.
also
Propositions
1.4,
(i);
1.7,
(v)];
Proposition
1.11,
(vi)).
To
verify
the
necessity
of
the
condition
of
assertion
(i),
observe
that
it
follows
formally
from
the
factorization
of
Definition
1.3,
(iv),
(a),
that
φ
is
either
a
morphism
of
Frobenius
type,
a
step,
or
a
pull-back
morphism.
Thus,
by
Propositions
1.7,
(v);
1.10,
(iv);
1.11,
(vi),
the
irreducibility
of
φ
implies
immediately
that
φ
is
a
morphism
as
in
(a),
(b),
or
(c).
42
SHINICHI
MOCHIZUKI
Next,
we
consider
assertion
(ii).
To
verify
the
sufficiency
of
the
condition
of
assertion
(ii),
observe
first
that
by
the
factorization
of
Definition
1.3,
(iv),
(a),
we
may
write
φ
=
α
◦
β
◦
γ,
where
α
is
a
pull-back
morphism,
β
is
a
pre-step,
and
γ
is
a
morphism
of
Frobenius
type.
By
assertion
(i)
[cf.
also
Proposition
1.10,
(v)],
it
follows
that
γ
is
an
isomorphism;
thus,
we
may
assume
without
loss
of
generality
that
γ
is
the
identity,
i.e.,
φ
=
α
◦
β.
On
the
other
hand,
it
follows
formally
from
the
fact
that
φ
is
an
FSM-morphism
that
α
is
fiberwise-surjective
[cf.
§0].
Next,
I
claim
that
α
is
a
monomorphism.
Indeed,
write
φ
:
A
→
B,
β
:
A
→
C,
α
:
C
→
B;
let
1
,
2
:
D
→
C
be
such
that
α
◦
1
=
α
◦
2
.
Then
by
Remark
1.1.1,
it
follows
immediately
that
deg
Fr
(
1
)
=
deg
Fr
(
2
),
Div(
1
)
=
Div(
2
),
hence,
by
applying
the
factorization
of
Definition
1.3,
(iv),
(a)
[and
the
total
epimorphicity
of
C;
cf.
also
Definition
1.3,
(ii),
and
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d)],
we
may
assume
without
loss
of
generality
[from
the
point
of
view
of
showing
that
α
is
a
monomorphism]
that
1
,
2
are
pull-back
morphisms.
Now
by
“adding
the
pull-backs
of
β
∗
(Div(β))
via
1
,
2
”
[cf.
Proposition
1.11,
(v);
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d)],
it
follows
that
there
exists
a
pre-step
ζ
:
E
→
D
such
that
there
exist
γ
1
,
γ
2
∈
Arr(C)
satisfying
1
◦
ζ
=
β
◦
γ
1
,
2
◦
ζ
=
β
◦
γ
2
.
Thus,
we
have:
φ
◦
γ
1
=
α
◦
β
◦
γ
1
=
α
◦
1
◦
ζ
=
α
◦
2
◦
ζ
=
α
◦
β
◦
γ
2
=
φ
◦
γ
2
.
But
since
φ
is
[an
FSM-morphism,
hence,
in
particular]
a
monomorphism,
it
follows
that
γ
1
=
γ
2
,
hence
[by
the
total
epimorphicity
of
C]
that
1
=
2
.
This
completes
the
proof
of
the
claim.
In
particular,
we
conclude
that
α
is
an
FSM-morphism.
Thus,
it
follows
[cf.
Proposition
1.11,
(vi)]
that
Base(α)
is
an
FSM-morphism
of
D.
Since,
however,
we
are
operating
under
the
assumption
that
D
is
of
FSMFF-
type,
it
follows
that
if
α
is
not
an
isomorphism,
then
Base(α)
admits
a
subordinate
[cf.
condition
(a)
of
the
definition
of
a
“category
of
FSMFF-type”
in
§0]
FSMI-
morphism,
which
implies
[cf.
Proposition
1.11,
(vi)]
that
α
admits
a
subordinate
FSMI-morphism
[which
is
also
a
pull-back
morphism].
Since
φ,
however,
is
as-
sumed
to
be
mid-adjoint
to
the
irreducible
morphisms
which
are
not
pre-steps,
we
thus
obtain
a
contradiction.
Thus,
α
is
an
isomorphism,
so
φ
is
a
pre-step.
This
completes
the
proof
of
the
sufficiency
of
the
condition
of
assertion
(ii).
Next,
we
consider
the
necessity
of
the
condition
of
assertion
(ii).
Thus,
suppose
that
φ
is
a
pre-step.
By
Proposition
1.11,
(vii),
φ
is
an
FSM-morphism;
by
Proposition
1.7,
(v),
φ
is
mid-adjoint
to
the
non-pre-steps.
This
completes
the
proof
of
assertion
(ii).
Next,
we
consider
assertion
(iii).
By
assertion
(i),
it
suffices
to
show
that
assertion
(iii)
holds
for
each
of
the
three
types
of
morphisms
“(a),
(b),
(c)”
discussed
in
assertion
(i).
If
φ
is
an
irreducible
pre-step,
then
it
follows
immediately
—
by
taking
ψ
to
be
a
prime-Frobenius
morphism
of
increasingly
large
Frobenius
degree
[cf.
Proposition
1.10,
(ii)]
—
that
the
condition
in
the
statement
of
assertion
(iii)
is
false
[as
desired].
On
the
other
hand,
if
φ
is
a
non-pre-step,
then
it
is
an
isometry.
Now
if
the
condition
in
the
statement
of
assertion
(iii)
is
false,
then
there
exist
equalities
α
n
◦
α
n−1
◦
.
.
.
◦
α
2
◦
α
1
=
ψ
◦
φ
where
α
1
,
.
.
.
,
α
n
,
ψ
are
FSMI-morphisms,
and
n
is
arbitrarily
large.
Here,
we
note
that
since
ψ
◦φ
and
ψ
are
FSM-morphisms,
it
thus
follows
formally
that
φ
is
also
an
FSM-morphism.
Next,
observe
that
since
φ
is
an
isometry,
it
follows
from
the
fact
THE
GEOMETRY
OF
FROBENIOIDS
I
43
that
ψ
is
irreducible
[cf.
also
assertion
(i);
Definition
1.1,
(ii),
(b);
Remark
1.1.1]
that
Div(ψ
◦
φ)
is
either
zero
or
irreducible;
since,
moreover,
deg
Fr
(ψ
◦
φ)
always
divides
a
product
of
two
prime
numbers
[cf.
assertion
(i);
the
irreducibility
of
φ,
ψ],
it
thus
follows
that
in
any
factorization
of
ψ
◦
φ
by
FSMI-morphisms,
all
but
three
[i.e.,
corresponding
to
two
possible
prime
factors
of
the
Frobenius
degree,
plus
one
possible
irreducible
factor
of
the
zero
divisor]
of
the
factorizing
FSMI-morphisms
are
pull-back
morphisms
[cf.
assertion
(i)].
On
the
other
hand,
this
implies
that
factorizations
of
arbitrarily
large
length
determine
chains
of
FSMI-morphisms
[cf.
assertion
(i);
Proposition
1.11,
(vi)]
originating
from
the
projection
to
D
of
the
domain
of
φ
which
are
also
of
arbitrarily
large
length,
a
contradiction
[cf.
condition
(b)
of
the
definition
of
a
“category
of
FSMFF-type”
in
§0].
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
Since
deg
Fr
(β)
=
deg
Fr
(β
),
it
follows
from
Remark
1.1.1
that
deg
Fr
(α)
=
deg
Fr
(α
),
hence
[since
α,
α
are
irreducible],
by
assertion
(i),
that
α
is
a
prime-Frobenius
morphism
if
and
only
if
α
is.
This
completes
the
proof
of
assertion
(iv).
Finally,
we
consider
assertion
(v).
First,
we
observe
that
the
necessity
of
the
condition
in
the
statement
of
assertion
(v)
[where
we
take
ψ
to
be
a
prime-Frobenius
morphism
such
that
deg
Fr
(φ)
=
deg
Fr
(ψ)]
follows
immediately
from
Proposition
1.10,
(i)
[cf.
also
Definition
1.3,
(ii);
assertion
(i);
the
first
equivalence
of
categories
of
Definition
1.3,
(iii),
(d)].
Next,
we
consider
sufficiency.
To
show
that
φ
is
a
prime-Frobenius
morphism,
it
suffices
[by
assertion
(i)]
to
show
that
it
is
not
a
pull-back
morphism.
Thus,
suppose
that
φ
is
a
pull-back
morphism.
Since
A
is
non-
group-like,
it
follows
[cf.
Proposition
1.4,
(iii)]
that
there
exists
a
step
α
:
A
→
B,
hence
that
there
exist
ψ,
β
as
in
the
statement
of
assertion
(v).
By
assertions
(i),
def
def
(iv),
ψ
is
also
a
pull-back
morphism.
Write
x
=
Div(α),
y
=
α
∗
(Div(β))
[where,
for
simplicity,
we
write
α
∗
for
Φ(Base(α))].
Then
by
Remark
1.1.1,
it
follows
that
φ
∗
(x
+
y)
=
x,
i.e.,
that
φ
∗
(x)
≤
x.
Since
x
=
0
is
arbitrary
[cf.
the
first
equivalence
of
categories
of
Definition
1.3,
(iii),
(d)],
it
thus
follows
from
our
assumption
that
Φ
is
non-dilating
that
φ
∗
is
the
identity
morphism.
But
this
implies
that
x
+
y
=
x,
i.e.,
[since
Φ
is
integral
—
cf.
Definition
1.1,
(i)]
that
y
=
0,
in
contradiction
to
our
assumption
that
β
is
a
step
[i.e.,
not
just
a
[necessarily
co-angular!]
pre-step
—
cf.
Proposition
1.4,
(i),
(iii)].
Thus,
we
conclude
[cf.
assertion
(iv)]
that
φ,
ψ
are
prime-Frobenius
morphisms,
of
the
same
Frobenius
degree.
In
particular,
if
def
def
we
write
x
=
Div(α),
y
=
α
∗
(Div(β)),
then
it
follows
[cf.
Remark
1.1.1]
that
φ
∗
(x
+
y)
=
deg
Fr
(φ)
·
x,
i.e.,
that
φ
∗
(x)
x
[cf.
§0],
hence
[by
our
assumption
that
Φ
is
non-dilating]
that
φ
∗
is
the
identity
morphism.
This
completes
the
proof
of
assertion
(v).
44
SHINICHI
MOCHIZUKI
Section
2:
Frobenius
Functors
In
the
present
§2,
we
discuss
various
functors
between
Frobenioids
that
are
intended
to
be
reminiscent
of
the
Frobenius
morphism
in
positive
characteristic
scheme
theory.
In
the
following
discussion,
we
maintain
the
notation
of
§1.
Also,
we
assume
that
we
have
been
given
a
divisorial
monoid
Φ
on
a
connected,
totally
epimorphic
category
D
and
a
Frobenioid
C
→
F
Φ
.
Proposition
2.1.
(The
Naive
Frobenius
Functor)
Let
d
∈
N
≥1
.
Then:
(i)
The
assignment
A
→
A
;
φ
→
φ
—
where
φ
:
A
→
B
is
an
arbitrary
morphism
of
C;
α
:
A
→
A
,
β
:
B
→
B
are
morphisms
of
Frobenius
type
of
Frobenius
degree
d;
φ
is
the
unique
morphism
such
that
φ
◦
α
=
β
◦
φ
[cf.
Proposition
1.10,
(i)]
—
determines
a
functor
Ψ:
C→C
[well-defined
up
to
isomorphism
of
functors]
which
we
shall
refer
to
as
the
naive
Frobenius
functor
[of
degree
d]
on
C.
Finally,
the
composite
of
the
naive
Frobe-
nius
functor
of
degree
d
1
∈
N
≥1
on
C
with
the
naive
Frobenius
functor
of
degree
d
2
∈
N
≥1
on
C
is
isomorphic
to
the
naive
Frobenius
functor
of
degree
d
1
·
d
2
on
C.
(ii)
The
functor
Ψ
of
(i)
is
“1-compatible”,
relative
to
C
→
F
Φ
,
with
the
functor
F
Φ
→
F
Φ
—
which
we
shall
refer
to
as
the
Frobenius
functor
on
F
Φ
—
determined
[cf.
Definition
1.1,
(iii)]
by
the
endomorphism
of
the
functor
Φ
given
by
multiplication
by
d.
Moreover,
if,
in
the
notation
of
(i),
A
=
A
,
A
is
Frobenius-normalized,
and
the
morphism
α
:
A
→
A
is
taken
to
be
a
base-
identity
endomorphism,
then
the
morphism
of
monoids
O
(A)
→
O
(A
)
induced
by
Ψ
is
given
by
raising
to
the
d-th
power.
(iii)
C
is
of
perfect
type
if
and
only
if
Ψ
is
an
equivalence
of
categories.
Proof.
Assertions
(i),
(ii)
follow
immediately
from
Definition
1.3,
(ii);
Proposition
1.10,
(i)
[cf.
also
Proposition
1.7,
(i)].
Finally,
we
consider
assertion
(iii).
The
sufficiency
of
the
condition
of
assertion
(iii)
follows
immediately
from
the
definition
of
“perfect”
[cf.
Definition
1.2,
(iv);
Remark
1.1.1].
To
verify
necessity,
suppose
that
C
is
of
perfect
type.
Then
the
essential
surjectivity
of
Ψ
follows
immediately
from
the
definition
of
“perfect”
[cf.
Definition
1.2,
(iv)].
To
verify
that
Ψ
is
fully
faithful,
we
reason
as
follows:
In
light
of
the
1-compatibility
of
Ψ
with
the
Frobenius
functor
on
F
Φ
[cf.
assertion
(ii)],
the
total
epimorphicity
of
C,
and
the
factorization
of
Definition
1.3,
(iv),
(a),
it
follows
immediately
that
one
may
reduce
to
the
case
of
linear
morphisms
by
applying
the
existence
and
(essential)
uniqueness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Definition
1.3,
(ii)].
Moreover,
THE
GEOMETRY
OF
FROBENIOIDS
I
45
by
applying
the
equivalence
of
categories
[involving
pull-backs]
of
Definition
1.3,
(i),
(c)
[cf.
also
the
isomorphism
of
functors
appearing
in
the
definition
of
a
“pull-back
morphism”
in
Definition
1.2,
(ii)],
one
may
reduce
further
to
the
case
of
pre-steps.
But
the
case
of
pre-steps
follows
immediately
from
the
definition
of
“perfect”
[cf.
Definition
1.2,
(iv)].
This
completes
the
proof
of
assertion
(iii).
Remark
2.1.1.
If
C
is
of
perfect
type,
then
for
any
d
=
a/b
∈
Q
>0
,
where
a,
b
∈
N
≥1
,
composing
the
naive
Frobenius
functor
of
degree
a
with
some
quasi-
inverse
functor
to
the
naive
Frobenius
functor
of
degree
b
yields
a
“naive
Frobenius
functor
of
degree
d”,
which,
by
Proposition
2.1,
(i),
is
independent
of
the
choice
of
a,
b.
Proposition
2.2.
(The
Functor
O
(−))
Write
D
∗
for
the
category
whose
objects
are
the
objects
of
C
istr
and
whose
morphisms
are
given
as
follows:
def
Hom
D
∗
(A,
B)
=
Hom
D
(A
D
,
B
D
)
def
def
[where
A,
B
∈
Ob(C
istr
);
A
D
=
Base(A);
B
D
=
Base(B)].
Thus,
the
natural
projection
functor
C
→
D
determines
natural
functors
C
istr
→
D
∗
→
D.
Moreover:
(i)
The
functor
D
∗
→
D
is
an
equivalence
of
categories.
(ii)
There
is
a
unique
contravariant
functor
D
∗
→
Mon
Ob(C
istr
)
=
Ob(D
∗
)
A
→
O
(A)
∈
Ob(Mon)
such
that
for
φ
:
A
→
B
in
Arr(C
istr
),
with
image
φ
D
∗
in
D
∗
,
the
following
properties
are
satisfied:
(a)
if
φ
is
a
[necessarily
co-angular
–
cf.
Proposition
1.4,
(i)]
linear
morphism,
then
O
(φ
D
∗
)
:
O
(B)
→
O
(A)
is
the
inclusion
of
Proposition
1.11,
(iv);
(b)
if
φ
is
a
[necessarily
co-angular]
pre-step,
then
O
(φ
D
∗
)
:
O
(B)
→
O
(A)
is
the
bijection
of
Definition
1.3,
(iii),
(c).
By
abuse
of
notation,
we
shall
also
denote
by
“O
(−)”
the
restriction
of
this
functor
on
D
∗
to
(C
istr
)
lin
.
Finally,
by
applying
the
equivalence
of
categories
of
(i),
we
obtain
a
contravariant
functor
D
→
Mon,
which,
by
abuse
of
notation,
we
shall
also
denote
by
“O
(−)”,
and
which
is
well-defined
up
to
isomorphism.
(iii)
The
assignment
Ob(C
istr
)
A
→
O
×
(A)
(⊆
O
(A))
determines
a
sub-
functor
of
the
functor
of
(ii)
which
is
equal
to
the
subfunctor
A
→
O
(A)
±
[cf.
the
notation
of
§0].
Moreover,
the
operation
“Div(−)”
determines
a
functorial
homomorphism
O
(A)
→
Φ(A)
that
induces
an
inclusion
O
(A)
char
=
O
(A)/O
×
(A)
→
Φ(A)
[cf.
the
notation
of
§0].
(iv)
If
φ
:
A
→
A
istr
is
an
isotropic
hull
in
C,
then
φ
determines
a
natural
inclusion
of
monoids
O
(A)
→
O
(A
istr
).
46
SHINICHI
MOCHIZUKI
Proof.
As
for
assertion
(i),
essential
surjectivity
follows
immediately
from
Defini-
tion
1.3,
(i),
(a)
[i.e.,
applied
to
the
Frobenioid
C
istr
—
cf.
Proposition
1.9,
(v)],
while
fully
faithfulness
follows
formally
from
the
definition
of
the
category
D
∗
.
Next,
def
def
we
consider
assertion
(ii).
Let
A,
B
∈
Ob(C
istr
);
A
D
=
Base(A);
B
D
=
Base(B).
Now
observe
that
any
morphism
A
D
→
B
D
in
D
factors
as
the
composite
of
an
∼
def
isomorphism
A
D
→
C
D
,
where
C
∈
Ob(C
istr
),
C
D
=
Base(C),
with
a
morphism
C
D
→
B
D
which
is
the
projection
to
D
of
a
pull-back
morphism
C
→
B
of
C
istr
[cf.
Definition
1.3,
(i),
(c)];
moreover,
this
pull-back
morphism
is
uniquely
determined,
istr
as
an
object
of
C
B
,
up
to
isomorphism
[cf.
the
isomorphism
of
functors
appearing
in
the
definition
of
a
“pull-back
morphism”
in
Definition
1.2,
(ii)].
Thus,
it
follows
that
to
construct
the
desired
functor
“O
(−)”
on
D
∗
,
it
suffices
to
construct,
for
∼
each
isomorphism
φ
D
:
A
D
→
B
D
,
a
bijection
O
(φ
D
)
:
O
(A)
→
O
(B)
which
is
compatible
with
composition
of
isomorphisms.
[Indeed,
once
one
constructs
“O
(−)”
in
this
fashion,
the
fact
that
this
“O
(−)”
is
compatible
with
composites
of
morphisms
of
D
∗
follows
immediately
from
the
manifest
functoriality
of
the
in-
clusion
of
Proposition
1.11,
(iv).]
This
may
be
done
by
using
co-angular
pre-steps
γ
A
:
C
→
A,
γ
B
:
C
→
B
such
that
φ
D
=
Base(γ
B
)
◦
Base(γ
A
)
−1
[cf.
Definition
1.3,
∼
∼
(i),
(b)]
and
the
bijections
O
(γ
A
)
:
O
(A)
→
O
(C),
O
(γ
B
)
:
O
(B)
→
O
(C)
determined
by
γ
A
,
γ
B
[cf.
Definition
1.3,
(iii),
(c)].
Note,
moreover,
that
the
result-
ing
bijection
O
(γ
A
)
−1
◦
O
(γ
B
)
is
independent
of
the
choice
of
γ
A
,
γ
B
.
[Indeed,
if
δ
A
:
D
→
A,
δ
B
:
D
→
B
satisfy
φ
D
=
Base(δ
B
)
◦
Base(δ
A
)
−1
,
then
there
exist
[cf.
Definition
1.3,
(i),
(b)]
co-angular
pre-steps
C
:
E
→
C,
D
:
E
→
D
such
that
Base(γ
A
)
◦
Base(
C
)
=
Base(δ
A
)
◦
Base(
D
)
—
which
[since
Base(δ
B
)
◦
Base(δ
A
)
−1
=
Base(γ
B
)
◦
Base(γ
A
)
−1
]
implies
that
Base(γ
B
)
◦
Base(
C
)
=
Base(δ
B
)
◦
Base(
D
)
hence
that
O
(γ
A
◦
C
)
=
O
(δ
A
◦
D
);
O
(γ
B
◦
C
)
=
O
(δ
B
◦
D
)
[cf.
Definition
1.3,
(iii),
(c)],
i.e.,
that
O
(
C
)
◦
O
(γ
A
)
=
O
(
D
)
◦
O
(δ
A
);
O
(
C
)
◦
O
(γ
B
)
=
O
(
D
)
◦
O
(δ
B
)
—
that
is
to
say
O
(γ
A
)
−1
◦
O
(γ
B
)
=
O
(δ
A
)
−1
◦
O
(δ
B
)
as
desired.]
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
is
immediate
from
the
definitions
[cf.
also
Definition
1.3,
(iii),
(b);
Definition
1.3,
(vi)].
Assertion
(iv)
follows
immediately
from
the
“universal
property
of
an
isotropic
hull”
[cf.
Definition
1.2,
(iv)]
and
the
fact
that
an
isotropic
hull
is
always
a
monomorphism
[cf.
Definition
1.3,
(v),
(a)].
THE
GEOMETRY
OF
FROBENIOIDS
I
Definition
2.3.
in
monoids
47
We
shall
refer
to
as
a
characteristic
splitting
on
C
a
subfunctor
τ
:
(C
istr
)
lin
→
Mon
of
the
functor
O
(−)
:
(C
istr
)
lin
→
Mon
of
Proposition
2.2,
(ii),
such
that
the
following
properties
hold:
(a)
for
every
A
∈
Ob(C
istr
),
τ
(A)
maps
bijectively
onto
O
(A)
char
,
hence
determines
a
splitting
of
monoids
∼
O
×
(A)
×
τ
(A)
→
O
(A)
which
is
functorial
in
A;
(b)
for
every
isotropic
hull
A
→
A
istr
of
C,
τ
(A
istr
)
⊆
O
(A
istr
)
lies
in
the
image
of
O
(A)
via
the
natural
injection
of
Proposition
2.2,
(iv).
Definition
2.4.
(i)
We
shall
say
that
M
∈
Ob(Mon)
is
perf-factorial
if
it
satisfies
the
following
conditions:
(a)
M
is
divisorial.
(b)
For
every
p
∈
Prime(M
)
[cf.
§0],
the
monoid
M
p
is
monoprime
[cf.
§0].
(c)
The
map
def
p∈Prime(M
)
rlf
M
pf
→
M
factor
=
M
p
rlf
a
→
(.
.
.
,
sup(Bound
p
Ë
{0}
(a)),
.
.
.
)
def
[where
we
write
M
p
rlf
=
M
p
pf
⊗R
≥0
;
we
refer
to
§0
for
more
on
the
notation
“M
pf
”,
“M
p
pf
”,
“⊗R
≥0
”;
the
“sup”
at
the
index
p
is
taken
in
M
p
rlf
]
is
a
well-defined
[i.e.,
the
various
Bound
p
Ë
{0}
(a)
⊆
M
p
rlf
are
bounded
subsets]
injective
homomorphism
of
monoids
whose
image
lies
in
p∈Prime(M
)
M
p
pf
,
hence
determines
an
injective
homomorphism
def
pf
=
M
pf
→
M
factor
M
p
pf
p∈Prime(M
)
which
we
shall
refer
to
as
the
factorization
homomorphism
of
M
pf
.
We
shall
often
use
the
factorization
homomorphism
to
regard
M
pf
as
a
sub-
pf
rlf
⊆
M
factor
.
monoid
of
M
factor
rlf
(d)
If
a
∈
M
factor
,
then
we
shall
write
Supp(a)
⊆
Prime(M
)
for
the
subset
of
p
for
which
the
component
of
a
at
p
is
nonzero
and
refer
to
Supp(a)
as
the
pf
support
of
a.
Then
the
submonoid
M
pf
⊆
M
factor
satisfies
the
following
pf
pf
property:
If
a
∈
M
factor
and
b
∈
M
satisfy
Supp(a)
⊆
Supp(b),
then
a
∈
M
pf
.
[Thus,
in
particular,
if
a,
b
∈
M
pf
,
then
an
inequality
“a
≤
b”
pf
.]
holds
in
M
pf
if
and
only
if
it
holds
in
M
factor
48
SHINICHI
MOCHIZUKI
Now
suppose
that
M
is
perf-factorial.
Then
we
shall
refer
to
the
[subset
which
is
easily
verified
to
be
a]
submonoid
rlf
M
rlf
⊆
M
factor
rlf
of
elements
a
∈
M
factor
such
that
there
exists
a
b
∈
M
pf
satisfying
Supp(a)
⊆
pf
Supp(b)
as
the
realification
of
M
.
Thus,
both
the
submonoid
M
pf
⊆
M
factor
and
the
rlf
rlf
submonoid
M
⊆
M
factor
are
completely
determined
by
the
collection
of
subsets
Supp(a)
⊆
Prime(M
),
as
a
ranges
over
the
elements
of
M
pf
;
if
a,
b
∈
M
rlf
,
then
an
rlf
;
inequality
“a
≤
b”
holds
in
M
rlf
if
and
only
if
it
holds
in
M
factor
rlf
)
gp
=
(M
rlf
)
gp
⊆
(M
factor
(M
p
rlf
)
gp
p∈Prime(M
)
is
an
R-vector
space.
Finally,
one
verifies
immediately
that
M
pf
,
M
rlf
are
also
perf-factorial.
(ii)
Let
Λ
be
a
monoid
type.
Then
we
shall
say
that
Λ
supports
M
∈
Ob(Mon)
if
any
of
the
following
conditions
hold:
(a)
Λ
=
Z;
(b)
Λ
=
Q,
and
M
is
perfect;
(c)
Λ
=
R,
M
is
perfect
and
perf-factorial,
and
for
every
p
∈
Prime(M
),
the
monoid
M
p
is
R-monoprime.
Note
that
if
Λ
supports
M
,
then
Λ
>0
acts
naturally
on
M
.
(iii)
Let
Λ
be
a
monoid
type
that
supports
Φ
[cf.
Definition
1.1,
(ii)];
d
∈
Λ
>0
.
Then
we
shall
write
d
·
Φ(−)
⊆
Φ(−)
for
the
subfunctor
of
Φ
determined
by
the
assignment
Ob((C
istr
)
lin
)
(Φ(A))
(⊆
Φ(A))
and
C
(d)
⊆
C
A
→
d
·
for
the
subcategory
determined
by
the
arrows
whose
zero
divisor
lies
in
d
·
Φ(−)
⊆
Φ(−).
Finally,
multiplication
by
d
on
Φ(−)
determines
a
“Frobenius
functor”
[as-
sociated
to
d
—
cf.
Proposition
2.1,
(ii)]
F
Φ
→
F
Φ
which
is
compatible
with
Frobenius
degrees
and
the
natural
projection
functor
F
Φ
→
D.
Proposition
2.5.
(The
Unit-linear
Frobenius
Functor)
Let
τ
be
a
charac-
teristic
splitting
on
C;
Λ
a
monoid
type
that
supports
Φ;
d
∈
Λ
>0
.
Suppose
that
the
Frobenioid
C
is
of
Frobenius-normalized,
metrically
trivial,
and
Aut-
ample
type.
Then:
(i)
The
natural
inclusion
O
(A)
char
→
Φ(A),
where
A
∈
Ob(C
istr
),
of
Propo-
sition
2.2,
(iii),
is,
in
fact,
a
bijection.
(ii)
C
istr
is
of
base-trivial
type.
Moreover,
every
object
of
C
istr
is
Frobenius-
trivial.
THE
GEOMETRY
OF
FROBENIOIDS
I
49
(iii)
There
exists
an
equivalence
of
categories
∼
Ψ
:
C
→
C
(d)
—
which
we
shall
refer
to
as
the
unit-linear
Frobenius
functor
[associated
to
τ
,
d]
—
that
satisfies
the
following
properties:
(a)
Ψ
acts
as
the
identity
on
objects
and
isometries
of
C;
(b)
Ψ
is
1-compatible,
relative
to
the
functors
C
→
F
Φ
,
C
(d)
→
F
d·Φ
=
(F
Φ
)
(d)
⊆
F
Φ
with
the
Frobenius
functor
associated
to
d
on
F
Φ
[which
implies,
in
particular,
that
C
(d)
,
equipped
with
the
natural
functor
C
(d)
→
F
d·Φ
,
is
a
Frobenioid].
Proof.
First,
we
observe
that,
by
applying
either
of
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d),
assertion
(i)
follows
formally
from
the
fact
that
C
is
of
metrically
trivial
and
Aut-ample
type.
Next,
we
consider
assertion
(ii).
Since
C
is
of
metrically
trivial
type,
it
follows
from
the
existence
of
[necessarily
co-angular
—
cf.
Proposition
1.4,
(i)]
pre-steps
relating
base-isomorphic
objects
of
C
istr
[cf.
Definition
1.3,
(i),
(b)],
that
the
isomorphism
class
of
an
object
of
C
istr
is
completely
determined
by
the
isomorphism
class
of
D
to
which
it
projects.
In
particular,
it
follows
from
the
existence
of
Frobenius-trivial
objects
[cf.
Definition
1.3,
(i),
(a)]
that
every
object
of
C
istr
is
Frobenius-trivial.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
consider
assertion
(iii).
By
applying
the
factorizations
of
Definition
1.3,
(iv),
(a);
(v),
(c),
together
with
the
bijection
of
assertion
(i)
[cf.
also
the
equiv-
alences
of
categories
of
Definition
1.3,
(iii),
(d)],
we
conclude
that
every
morphism
φ
of
C
admits
a
factorization
φ
=
α
◦
β
◦
γ
◦
δ
in
C,
where
α
is
a
pull-back
morphism;
β
is
a
base-identity
pre-step
endomorphism
[hence
is
co-angular,
by
Definition
1.3,
(iii),
(b)];
γ
is
an
isometric
pre-step;
δ
is
a
morphism
of
Frobenius
type.
Moreover,
this
factorization
is
unique
[cf.
Definition
1.3,
(iv),
(a);
(v),
(c)],
up
to
replacing
(α,
β,
γ,
δ)
by
(α◦
,
−1
◦β◦ζ,
ζ
−1
◦γ◦θ,
θ
−1
◦δ),
where
,
θ
are
isomorphisms
of
C,
and
ζ
=
β
◦
,
for
some
base-identity
automorphism
β
.
Suppose
that
β
∈
O
(A),
for
A
∈
Ob(C).
Thus,
by
applying
the
characteristic
splitting
∼
O
×
(A)
×
τ
(A)
→
O
(A)
[which
applies
even
if
A
is
not
isotropic
—
cf.
Definition
2.3,
(a),
(b)]
to
β
∈
O
(A),
we
obtain
a
factorization
β
=
β
0
·
β
1
[where
β
0
∈
O
×
(A),
β
1
∈
τ
(A)].
Now
we
set
def
Ψ(β)
=
β
0
·
β
1
d
;
def
Ψ(φ)
=
α
◦
Ψ(β)
◦
γ
◦
δ
50
SHINICHI
MOCHIZUKI
[where
we
note
that
the
expression
“β
1
d
”
makes
sense
for
d
∈
Λ
>0
,
by
assertion
(i);
Definition
2.4,
(ii)].
Then
it
follows
immediately
from
the
functoriality
of
the
∼
characteristic
splitting
τ
(−)
that
for
any
isomorphism
:
A
→
A
in
C,
β
∈
O
×
(A),
we
have
Ψ(
−1
◦
β
◦
β
◦
)
=
−1
◦
Ψ(β)
◦
β
◦
.
This
implies
immediately
that
Ψ(φ)
is
independent
of
the
choice
of
factorization
φ
=
α
◦
β
◦
γ
◦
δ.
Next,
observe
that
by
assertion
(ii),
it
follows
that
if
φ
∈
Arr(C
istr
),
then
the
morphism
of
Frobenius
type
δ
may
be
taken
to
be
a
base-identity
endomorphism.
Thus,
by
the
functoriality
of
τ
with
respect
to
morphisms
of
(C
istr
)
lin
,
and
our
assumption
that
C
is
of
Frobenius-normalized
type
—
together
with
the
elementary
computation
Ψ(β
d
)
=
Ψ(β
0
d
·
β
1
d
)
=
β
0
d
·
(β
1
d
)
d
=
(β
0
·
β
1
d
)
d
=
Ψ(β)
d
for
d
∈
N
≥1
—
it
follows
that
the
assignment
φ
→
Ψ(φ)
is
compatible
with
com-
posites,
at
least
when
φ
∈
Arr(C
istr
).
On
the
other
hand,
since
isotropic
hulls
are
monomorphisms
[cf.
Definition
1.3,
(v),
(a)],
this
implies
[by
relating
an
arbitrary
φ
∈
Arr(C)
to
the
result
of
applying
the
isotropification
functor
of
Proposition
1.9,
(v),
to
φ]
that
the
assignment
φ
→
Ψ(φ)
is
compatible
with
composites,
for
arbitrary
φ
∈
Arr(C).
This
completes
the
definition
of
a
functor
Ψ
:
C
→
C
(d)
which
satisfies
the
properties
(a),
(b)
in
the
statement
of
Proposition
2.5,
(iii).
On
the
other
hand,
it
is
clear
from
the
definition
of
Ψ,
C
(d)
that
Ψ
is
essentially
surjective,
faithful,
and
full
[cf.
assertion
(i)].
This
completes
the
proof
of
Proposition
2.5.
Remark
2.5.1.
If
C
is
of
isotropic
and
unit-trivial
type,
then
the
“unit-linear
Frobenius
functor”
of
Proposition
2.5,
(iii),
may
be
regarded
as
a
sort
of
gener-
alization
of
the
“naive
Frobenius
functor”
of
Proposition
2.1,
(i),
to
the
case
of
d
∈
N
≥1
.
Corollary
2.6.
(Unit-wise
Frobenius
Functors
I)
Let
τ
be
a
character-
istic
splitting
on
C;
d
∈
N
≥1
.
Suppose
that
the
Frobenioid
C
is
of
Frobenius-
normalized,
metrically
trivial,
and
Aut-ample
type.
Then
there
exists
a
func-
tor
Ψ:
C→C
—
which
we
shall
refer
to
as
the
unit-wise
Frobenius
functor
[associated
to
τ
,
d]
—
which
satisfies
the
following
properties:
(a)
Ψ
is
1-compatible,
relative
to
the
functor
C
→
F
Φ
,
with
the
identity
functor
on
F
Φ
.
(b)
Ψ
maps
an
object
(respectively,
morphism
of
Frobenius
type;
pre-step;
pull-
back
morphism)
of
C
istr
to
an
isomorphic
object
(respectively,
abstractly
equiva-
lent
morphism;
abstractly
equivalent
morphism;
abstractly
equivalent
mor-
phism)
of
C.
(c)
If
A
∈
Ob(C
istr
),
then
there
exists
an
isomorphism
Ψ(A)
∼
=
A
such
that
the
endomorphism
of
O
×
(A)
induced
by
Ψ
followed
by
conjugation
by
this
isomorphism
is
given
by
raising
to
the
d-th
power.
THE
GEOMETRY
OF
FROBENIOIDS
I
51
(d)
If
C
is
of
perfect
type,
then
Ψ
is
an
equivalence
of
categories.
If
d
=
1
or
C
is
of
isotropic
and
unit-trivial
type,
then
Ψ
is
isomorphic
to
the
identity
functor.
Proof.
First,
let
us
observe
that
the
naive
Frobenius
functor
C
→
C
associated
to
d
[cf.
Proposition
2.1,
(i)]
factors
naturally
through
the
subcategory
C
(d)
⊆
C
[cf.
Proposition
2.1,
(ii);
Definition
2.4,
(iii)];
write
Ψ
1
:
C
→
C
(d)
for
the
resulting
functor.
Next,
let
us
write
Ψ
2
:
C
(d)
→
C
for
some
quasi-inverse
functor
to
the
unit-linear
Frobenius
functor
[which
is
an
equivalence
of
categories]
associated
def
to
d
[cf.
Proposition
2.5,
(iii)].
Set
Ψ
=
Ψ
2
◦
Ψ
1
:
C
→
C.
Then
it
follows
immediately
from
Propositions
2.1,
(ii);
2.5,
(iii),
(b),
that
Ψ
satisfies
property
(a).
Since
[cf.
Proposition
2.5,
(ii)]
the
isomorphism
class
of
an
object
of
C
istr
is
completely
determined
by
the
isomorphism
class
of
D
to
which
it
projects,
it
thus
follows
that
Ψ
preserves
isomorphism
classes
of
objects
of
C
istr
.
Now
the
remainder
of
properties
(b),
(c),
(d)
follows
immediately
from
the
construction
of
Ψ
1
,
Ψ
2
in
the
proofs
of
Propositions
2.1,
(i);
2.5,
(iii)
[cf.
also
Remark
2.5.1;
Proposition
1.10,
(i);
Definition
1.3,
(ii);
equivalences
of
categories
of
Definition
1.3,
(iii),
(d);
the
definition
of
a
“pull-back
morphism”
in
Definition
1.2,
(ii);
Proposition
2.1,
(iii)].
Definition
2.7.
Suppose
that
the
Frobenioid
C
is
of
isotropic
type.
(i)
We
shall
refer
to
as
a
base-section
of
the
Frobenioid
C
any
subcategory
P
⊆
C
pl-bk
⊆
C
[where
C
pl-bk
⊆
C
is
as
in
Definition
1.3,
(i),
(c)]
satisfying
the
following
conditions:
(a)
P
is
a
skeleton
[cf.
§0];
(b)
every
object
of
P
is
Frobenius-
trivial;
(c)
the
composite
P
→
D
of
the
inclusion
functor
P
→
C
with
the
natural
projection
functor
C
→
D
is
an
equivalence
of
categories.
In
this
situation,
we
shall
refer
to
the
morphisms
of
C
that
lie
in
P
as
P-distinguished.
(ii)
Let
P
⊆
C
be
a
base-section.
Observe
that
since
D,
hence
also
P,
is
a
connected
category,
it
follows
immediately
that
for
any
∈
End(P
→
C),
it
makes
sense
to
speak
of
the
Frobenius
degree
deg
Fr
(
)
∈
N
≥1
of
—
i.e.,
the
Frobenius
degree
of
the
endomorphisms
in
C
[of
objects
of
P]
determined
by
[which,
since
P
is
connected,
is
easily
seen
to
be
independent
of
the
choice
of
object
of
P
—
cf.
Remark
1.1.1].
We
shall
refer
to
as
a
[P-]Frobenius-section
of
the
Frobenioid
C
any
homomorphism
of
monoids
F
:
N
≥1
→
End(P
→
C)
satisfying
the
following
conditions:
(a)
the
composite
of
F
with
the
homomorphism
End(P
→
C)
→
N
≥1
determined
by
considering
the
Frobenius
degree
is
the
iden-
tity
on
N
≥1
;
(b)
the
endomorphisms
of
objects
of
C
determined
by
an
element
of
End(P
→
C)
in
the
image
of
F
are
base-identity
endomorphisms
of
Frobenius
type.
We
shall
refer
to
a
Frobenius-section
F
which
is
regarded
as
being
known
only
up
to
composition
with
automorphisms
of
the
monoid
N
≥1
as
a
quasi-Frobenius-section.
If
F
is
a
Frobenius-section,
then
we
shall
refer
to
the
endomorphisms
of
C
induced
by
elements
of
End(P
→
C)
in
the
image
of
F
as
F
-distinguished.
52
SHINICHI
MOCHIZUKI
(iii)
We
shall
refer
to
a
pair
(P,
F
),
where
P
is
a
base-section
of
C,
and
F
is
a
P-Frobenius-section
of
C,
as
a
base-Frobenius
pair
of
C;
when
F
is
regarded
as
being
known
only
up
to
composition
with
automorphisms
of
the
monoid
N
≥1
,
we
shall
refer
to
such
a
pair
as
a
quasi-base-Frobenius
pair.
If
the
Frobenioid
C
admits
a
base-Frobenius
pair
[or,
equivalently,
a
quasi-base-Frobenius
pair],
then
we
shall
say
that
C
is
of
pre-model
type.
Remark
2.7.1.
The
notions
of
a
“base-section”
and
“Frobenius-section”
are
intended
to
be
a
sort
of
“category-theoretic
translation”
of
the
notion
of
a
“choice
of
trivialization
of
a
trivial
line
bundle”,
which
occurs
naturally
when
C
is
a
category
of
trivial
line
bundles
[cf.
Remark
5.6.1;
Examples
6.1,
6.3
below].
Remark
2.7.2.
Suppose
that
C
is
of
isotropic
type.
Let
(P,
F
)
be
a
base-Frobenius
pair
of
C.
Then
the
only
arrows
of
C
which
are
both
F
-
and
P-distinguished
[hence
base-identity
automorphisms
—
cf.,
e.g.,
the
factorization
of
Definition
1.3,
(iv),
(a)]
are
the
identity
arrows.
Suppose
further
that
the
Frobenioid
C
is
of
base-trivial
type,
and
that
the
category
C
is
a
skeleton.
Then
every
morphism
φ
of
C
admits
a
factorization
φ
=
α
◦
β
◦
γ
where
α
is
P-distinguished;
β
is
a
base-identity
pre-step
endomorphism;
γ
is
F
-
distinguished.
Moreover,
this
factorization
is
unique
[in
the
strict
sense
—
i.e.,
not
up
to
isomorphisms,
etc.].
[Indeed,
the
existence
and
uniqueness
of
the
factorization
in
question
follow
immediately
from
Definition
1.3,
(iv),
(a);
the
definition
of
P-,
F
-distinguished;
our
assumptions
concerning
C;
the
total
epimorphicity
of
C;
the
isomorphism
of
functors
appearing
in
the
definition
of
a
“pull-back
morphism”
in
Definition
1.2,
(ii).]
Definition
2.8.
(i)
If,
for
every
A
∈
Ob(C),
it
holds
that
O
×
(A)
admits
a
[uniquely
determined]
profinite
topology
such
that
O
×
(A),
equipped
with
this
topology,
is
a
topologically
finitely
generated
profinite
[abelian]
group,
then
we
shall
say
that
C
is
of
unit-
profinite
type.
(ii)
Suppose
that
M
is
a
topologically
finitely
generated
profinite
abelian
group.
Thus,
M
decomposes
as
a
direct
product
of
pro-l
groups
M
[l],
where
l
varies
over
the
elements
of
Primes
[cf.
§0].
We
shall
refer
to
the
factor
M
[l]
as
the
pro-l
portion
of
M
.
(iii)
Let
M
be
as
in
(ii);
assume
that
the
group
law
of
M
is
written
multiplica-
tively.
If
ζ
:
Primes
→
N
≥1
is
a
set-theoretic
function,
then
we
shall
refer
to
as
the
map
given
by
raising
to
the
ζ-th
power
on
M
the
map
M
→
M
(M
)
a
→
a
ζ
(∈
M
)
given
by
raising
to
the
ζ(l)-th
power
on
M
[l],
for
l
∈
Primes.
We
shall
refer
to
a
set-theoretic
function
ζ
:
Primes
→
N
≥1
as
being
of
co-prime
type
if
ζ
maps
each
THE
GEOMETRY
OF
FROBENIOIDS
I
53
element
l
∈
Primes
to
an
element
of
N
≥1
that
is
prime
to
l.
[Thus,
if
ζ
is
of
co-prime
type,
then
the
map
given
by
raising
to
the
ζ-th
power
will
always
be
bijective.]
Proposition
2.9.
(Unit-wise
Frobenius
Functors
II)
Suppose
that
the
Frobenioid
C
is
of
Frobenius-normalized,
base-trivial,
isotropic,
and
Aut-
ample
[cf.
Remark
2.9.2
below]
type.
Then:
(i)
If
the
base
category
D
admits
a
terminal
object
[cf.
§0],
then
C
is
of
pre-model
type.
(ii)
Let
τ
be
a
characteristic
splitting
on
C;
ζ
:
Primes
→
N
≥1
a
set-theoretic
function.
Suppose
that
C
is
of
pre-model
and
unit-profinite
type.
Then
there
exists
a
functor
Ψ:
C→C
—
which
we
shall
refer
to
as
the
unit-wise
Frobenius
functor
[associated
to
τ
,
ζ]
—
which
satisfies
the
following
properties:
(a)
Ψ
is
1-compatible,
relative
to
the
functor
C
→
F
Φ
,
with
the
identity
functor
on
F
Φ
.
(b)
Ψ
maps
an
object
(respectively,
morphism
of
Frobenius
type;
pre-step;
pull-back
morphism)
of
C
to
an
isomorphic
object
(respectively,
abstractly
equivalent
morphism;
abstractly
equivalent
morphism;
abstractly
equivalent
morphism)
of
C.
(c)
If
A
∈
Ob(C),
then
there
exists
an
isomorphism
Ψ(A)
∼
=
A
such
that
the
endomorphism
of
O
×
(A)
induced
by
Ψ
followed
by
conjugation
by
this
isomorphism
is
given
by
raising
to
the
ζ-th
power.
(d)
If
ζ
is
of
co-prime
type
[cf.
Definition
2.8,
(iii)],
then
Ψ
is
an
equiv-
alence
of
categories.
If
C
is
of
unit-trivial
type,
then
Ψ
is
isomorphic
to
the
identity
functor.
Proof.
By
well-known
general
nonsense
in
category
theory,
we
may
assume,
with-
out
loss
of
generality,
for
the
remainder
of
the
proof
of
Proposition
2.9,
that
the
category
C
is
a
skeleton.
Thus,
C
pl-bk
is
also
a
skeleton.
Now
we
consider
assertion
(i).
Observe
[cf.
Definition
1.3,
(i),
(c);
the
fact
that
C
is
of
base-trivial
type]
that
if
A
∈
Ob(C)
projects
to
a
terminal
object
of
D,
then
A
is
pseudo-terminal
[cf.
§0].
Note
that
by
Definition
1.3,
(i),
(a),
and
our
assumptions
on
D,
it
follows
that
such
an
object
A
always
exists;
let
us
fix
one
such
object
A.
Thus,
the
natural
projection
functor
determines
an
equivalence
of
categories
∼
pl-bk
C
A
→D
[cf.
Definition
1.3,
(i),
(c)].
Note
that
it
follows
immediately
from
the
existence
of
this
equivalence
of
categories
[together
with
the
fact
that
A
maps
to
a
terminal
54
SHINICHI
MOCHIZUKI
pl-bk
object
of
D]
that
the
natural
functor
C
A
→
C
pl-bk
is
injective
on
isomorphism
pl-bk
is
a
skeletal
subcategory
[cf.
§0],
then
classes
of
objects.
In
particular,
if
Q
⊆
C
A
[relative
to
some
sufficiently
large
universe
with
respect
to
which,
say,
the
category
C
is
small]
the
natural
map
Ob(Q)
→
Ob(C
pl-bk
)
=
Ob(C)
∼
pl-bk
is
bijective
[cf.
the
fact
that
C
is
of
base-trivial
type;
the
equivalence
C
A
→
D].
Thus,
the
subcategory
P
⊆
C
determined
by
the
image
of
the
objects
and
arrows
of
Q
in
C
is
a
skeleton
which
satisfies
the
conditions
of
Definition
2.7,
(i)
[cf.
the
fact
that
C
is
of
base-trivial
and
isotropic
type;
Definition
1.3,
(i),
(a)]
—
that
is
to
say,
P
is
a
base-section.
Next,
let
us
observe
[cf.
the
fact
that
C
is
of
base-trivial
and
isotropic
type;
Def-
inition
1.3,
(i),
(a)]
that
A
is
Frobenius-trivial,
hence
that
there
exists
a
morphism
of
monoids
F
A
:
N
≥1
→
End
C
(A)
whose
composite
with
the
morphism
of
monoids
deg
Fr
(−)
:
End
C
(A)
→
N
≥1
is
the
identity
morphism
on
N
≥1
,
and
whose
image
consists
of
base-identity
endomor-
phisms
of
Frobenius
type
of
A.
Thus,
by
Proposition
1.11,
(iii),
we
conclude
that
F
A
extends
to
a
P-Frobenius-section
F
:
N
≥1
→
End(P
→
C)
—
hence
that
C
is
of
pre-model
type,
as
desired.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
Observe
[cf.
the
fact
that
C
is
of
base-trivial
and
isotropic
type;
the
fact
that
C
is
a
skeleton]
we
may
apply
Remark
2.7.2
to
conclude
that
every
morphism
φ
:
C
→
D
of
C
admits
a
unique
factorization
φ
=
α
◦
β
◦
γ
in
C,
where
α
is
P-distinguished;
β
is
a
base-identity
pre-step
endomorphism;
γ
is
F
-distinguished.
Now
[cf.
the
proof
of
Proposition
2.5,
(iii)]
by
applying
the
∼
characteristic
splitting
[cf.
Definition
2.3,
(a)]
O
×
(C)
×
τ
(C)
→
O
(C),
we
may
write
β
=
β
0
·
β
1
∈
O
(C)
[where
β
0
∈
O
×
(C),
β
1
∈
τ
(C)].
Set
def
Ψ(β)
=
β
0
ζ
·
β
1
;
def
Ψ(φ)
=
α
◦
Ψ(β)
◦
γ
[where
“(−)
ζ
”
is
as
defined
in
Definition
2.8,
(iii)].
Since
β
is
completely
determined
by
φ,
it
follows
that
Ψ
is
well-defined
[as
a
“map
on
arrows”].
Moreover,
it
follows
from
the
definition
of
P-
and
F
-distinguished
morphisms
[together
with
the
fact
that
raising
to
the
ζ-th
power
defines
an
endomorphism
of
the
functor
in
monoids
“O
×
(−)”
on
C
lin
which
commutes
with
raising
to
the
d-th
power,
for
d
∈
N
≥1
—
cf.
THE
GEOMETRY
OF
FROBENIOIDS
I
55
our
assumption
that
C
is
of
Frobenius-normalized
type]
that
Ψ
is,
in
fact,
a
functor,
and
that
Ψ
satisfies
properties
(a),
(b),
(c),
(d)
in
the
statement
of
Proposition
2.9,
(ii).
This
completes
the
proof
of
Proposition
2.9.
Remark
2.9.1.
By
“base-changing”
the
Frobenioid
C
via
various
functors
D
→
D
as
in
Proposition
1.6,
it
follows
that
one
may
obtain
“unit-wise
Frobenius
functors”
as
in
Proposition
2.9,
(ii),
for
many
Frobenioids
whose
base
categories
do
not
nec-
essarily
admit
terminal
objects
[as
is
required
in
the
hypotheses
of
Proposition
2.9,
(i)].
Remark
2.9.2.
We
shall
see
later
[cf.
Theorem
5.1,
(iii)]
that
in
fact,
the
Aut-ampleness
hypothesis
in
the
statement
of
Proposition
2.9
is
superfluous.
56
SHINICHI
MOCHIZUKI
Section
3:
Category-theoreticity
of
the
Base
and
Frobenius
Degree
In
the
present
§3,
we
show
various
results
in
the
“opposite
direction”
to
the
direction
represented
by
the
various
Frobenius
functors
of
§2.
Namely,
we
show
that
various
natural
structures
—
such
as
Frobenius
degrees
and
the
natural
projection
functor
to
the
base
category
—
are
preserved
by
equivalences
of
categories
between
Frobenioids.
In
the
following
discussion,
we
maintain
the
notation
of
§1,
§2.
Also,
we
assume
that
we
have
been
given
a
divisorial
monoid
Φ
on
a
connected,
totally
epimorphic
category
D
and
a
Frobenioid
C
→
F
Φ
.
Definition
3.1.
(i)
We
shall
say
that
C
is
of
quasi-isotropic
type
if
it
holds
that
A
∈
Ob(C)
is
non-isotropic
if
and
only
if
it
is
an
iso-subanchor
[cf.
§0].
[Thus,
if
C
is
of
isotropic
type,
then
C
is
of
quasi-isotropic
type
—
cf.
Remark
3.1.1
below.]
We
shall
say
that
C
is
of
standard
type
if
the
following
conditions
are
satisfied:
(a)
C
is
of
quasi-isotropic
and
Frobenius-isotropic
type;
(b)
if
C
is
of
group-like
type,
then
C
istr
admits
a
Frobenius-compact
object;
(c)
C
is
of
Frobenius-normalized
type;
(d)
D
is
of
FSMFF-type;
(e)
Φ
is
non-dilating.
We
shall
say
that
a
category
E
is
Frobenius-slim
if
every
homomorphism
of
monoids
F
→
Aut(E
A
→
E)
[where
A
∈
Ob(E)]
factors
through
the
natural
surjection
F
N
≥1
.
[Thus,
every
slim
category
is
Frobenius-slim.]
(ii)
Write
C
Fr-tp
⊆
C
for
the
subcategory
of
C
determined
by
the
morphisms
of
Frobenius
type;
C
bi-Fr
⊆
C
Fr-tp
×
C
Fr-tp
for
the
subcategory
of
the
product
category
C
Fr-tp
×
C
Fr-tp
determined
by
pairs
of
morphisms
of
Frobenius
type
of
the
same
Frobenius
degree.
For
A,
B
∈
Ob(C),
we
shall
write
def
Hom
pf
C
(A,
B)
=
lim
−
→
(A→A
,B→B
)∈Ob(
(A,B)
C
bi-Fr
)
Hom
C
(A
,
B
)
where
the
inductive
limit
is
parametrized
by
[say,
some
small
skeletal
subcategory
of]
(A,B)
C
bi-Fr
;
the
map
Hom
C
(A
,
B
)
→
Hom
C
(A
,
B
)
induced
by
a
morphism
(A
→
A
,
B
→
B
)
in
(A,B)
C
bi-Fr
from
an
object
(A
→
A
,
B
→
B
)
of
(A,B)
C
bi-Fr
to
an
object
(A
→
A
,
B
→
B
)
of
(A,B)
C
bi-Fr
is
the
map
determined
by
the
assignment
“φ
→
φ
”
of
Proposition
1.10,
(i).
We
shall
refer
to
an
element
of
Hom
pf
C
(A,
B)
as
a
perfected
morphism
A
→
B.
(iii)
Suppose
that
the
Frobenioid
C
is
of
Frobenius-isotropic
type.
Then
we
shall
write
C
pf
THE
GEOMETRY
OF
FROBENIOIDS
I
57
for
the
category
—
which
we
shall
refer
to
as
the
perfection
of
C
—
defined
as
follows:
The
objects
of
C
pf
are
pairs
(A,
n),
where
A
∈
Ob(C),
n
∈
N
≥1
.
The
morphisms
of
C
pf
are
given
by
def
Hom
C
pf
((A,
n),
(B,
m))
=
Hom
pf
C
(A
,
B
)
where
(A,
n)
and
(B,
m)
are
objects
of
C
pf
;
A
→
A
is
a
morphism
of
Frobenius
type
in
C
of
Frobenius
degree
m;
B
→
B
is
a
morphism
of
Frobenius
type
in
C
of
Frobenius
degree
n;
one
verifies
immediately
[cf.
Definition
1.3,
(ii)]
that
this
set
of
morphisms
of
C
pf
from
(A,
n)
to
(B,
m)
is
independent
[up
to
uniquely
determined
natural
bijections]
of
the
choice
of
morphisms
of
Frobenius
type
A
→
A
,
B
→
B
;
composition
of
morphisms
of
C
pf
is
defined
in
the
evident
fashion.
[Thus,
in
words,
the
pair
(A,
n)
is
to
be
thought
of
as
an
“n-th
root
of
A”.]
Also,
we
obtain
a
natural
functor
C
→
C
pf
[by
mapping
“A
→
(A,
1)”].
(iv)
Two
co-objective
[cf.
§0]
morphisms
α
1
,
α
2
:
A
→
B
of
C
istr
will
be
called
unit-equivalent
if
there
exist
morphisms
γ
:
A
→
C,
β
:
C
→
B
[in
C
istr
]
and
an
automorphism
δ
∈
O
×
(C)
such
that
α
1
=
β
◦
γ,
α
2
=
β
◦
δ
◦
γ.
In
this
situation,
O
×
we
shall
write
α
1
≈
α
2
.
[Thus,
if
C
is
of
unit-trivial
type,
then
two
co-objective
morphisms
of
C
istr
are
unit-equivalent
if
and
only
if
they
are
equal.]
By
Proposition
O
×
3.3,
(ii),
below,
it
follows
that
“
≈
”
determines
an
equivalence
relation
on
the
set
of
morphisms
A
→
B
in
C
istr
which
is,
moreover,
closed
under
composition
of
morphisms;
we
shall
write
Hom
un-tr
C
istr
(A,
B)
for
the
set
of
unit-equivalence
classes
of
morphisms
A
→
B.
Also,
we
shall
write
C
un-tr
for
the
category
whose
objects
are
the
objects
of
C
istr
,
and
whose
morphisms
are
un-tr
as
the
unit-trivialization
of
C.
given
by
“Hom
un-tr
C
istr
(−,
−)”,
and
refer
to
C
Remark
3.1.1.
Observe
that:
An
iso-subanchor
of
the
Frobenioid
C
is
never
isotropic.
[In
particular,
if
C
is
of
isotropic
type,
then
C
is
of
quasi-isotropic
type.]
Indeed,
by
Proposition
1.10,
(iv),
an
anchor
is
never
isotropic.
Thus,
by
Definition
1.3,
(vii),
(b),
a
subanchor
is
never
isotropic.
Now
if
B
→
A
is
a
mono-minimal
categorical
quotient
[cf.
§0]
in
C
of
B
by
a
group
G
⊆
Aut
C
(B)
such
that
B
is
a
subanchor
and
A
is
isotropic,
then
applying
the
isotropification
functor
of
Proposition
1.9,
(v),
yields
a
factorization
B
→
B
→
A,
where
B
→
B
is
an
isotropic
hull
[hence
a
monomorphism
—
cf.
Definition
1.3,
(v),
(a)],
such
that
G
acts
compatibly
[relative
to
the
arrow
B
→
B
]
on
B
;
thus,
by
the
definition
of
the
term
“mono-minimal”
it
follows
that
the
arrow
B
→
B
is
an
isomorphism,
i.e.,
that
B
is
isotropic
—
a
contradiction.
This
completes
the
proof
of
the
“observation”.
58
SHINICHI
MOCHIZUKI
Remark
3.1.2.
Observe
that
for
any
residually
finite
group
G
[i.e.,
a
group
G
such
that
the
intersection
of
the
normal
subgroups
of
finite
index
of
G
is
trivial]:
Any
homomorphism
of
monoids
F
→
G
factors
through
the
natural
sur-
jection
F
N
≥1
.
[Indeed,
it
suffices
to
show
this
when
G
is
finite.
When
G
is
finite,
it
follows
immediately
from
the
definition
of
F
[cf.
Definition
1.1,
(iii)]
that
the
image
of
1
∈
Z
≥0
in
G
is
an
element
γ
∈
G
such
that
for
every
d
∈
N
≥1
,
there
exists
an
element
δ
d
∈
G
such
that
δ
d
·γ
·δ
d
−1
=
γ
d
.
Thus,
by
taking
d
to
be
the
order
of
γ,
we
conclude
that
γ
is
the
identity,
hence
that
the
homomorphism
of
monoids
F
→
G
factors
through
the
natural
surjection
F
N
≥1
,
as
desired.]
In
particular,
it
follows
that
if
E
is
a
category
such
that
for
every
A
∈
Ob(E),
the
group
Aut(E
A
→
E)
is
residually
finite,
then
E
is
Frobenius-slim.
Remark
3.1.3.
The
phenomenon
discussed
in
Remark
3.1.2
may
be
regarded
as
an
example
of
the
following
fundamental
dichotomy
[which
is,
in
a
certain
sense,
a
central
theme
of
the
theory
of
the
present
paper]
between
the
structure
of
the
base
category
D
and
the
“Frobenius
structure”
constituted
by
N
≥1
:
base
category
“indifferent
to
order”
groups
Frobenius
“order-conscious”
non-group-like
monoids
This
sort
of
phenomenon
may
be
observed
in
“classical
scheme
theory”
for
instance
in
the
invariance
of
the
étale
site
of
a
scheme
in
positive
characteristic
under
the
Frobenius
morphism.
Here,
it
is
useful
to
recall
that
a
typical
example
of
a
“base
category”
is
constituted
by
the
subcategory
of
connected
objects
of
a
Galois
category
[which
is
easily
verified
to
be
of
FSM-,
hence
also
of
FSMFF-type].
By
contrast,
categories
such
as
Order(Z
≥0
),
Order(N
≥1
)
or
[the
one-object
categories
determined
by]
Z
≥0
,
N
≥1
are
not
of
FSMFF-type.
In
this
context,
it
is
interesting
to
note
that
categories
such
as
Order(−)
of
a
finite
subset
of
Z
≥0
of
cardinality
≥
2
[with
the
induced
ordering]
constitute
a
sort
of
“borderline
case”,
which
is
of
FSMFF-,
but
not
of
FSM-,
type.
Proposition
3.2.
(Perfections
of
Frobenioids)
Suppose
that
the
Frobenioid
C
is
of
Frobenius-isotropic
type.
Then:
(i)
There
is
a
natural
1-commutative
diagram
of
functors
C
⏐
⏐
−→
C
pf
⏐
⏐
F
Φ
−→
F
Φ
pf
—
where
the
vertical
arrow
on
the
left
is
the
functor
that
defines
the
Frobenioid
structure
on
C;
the
vertical
arrow
on
the
right
is
induced
by
the
vertical
arrow
on
THE
GEOMETRY
OF
FROBENIOIDS
I
59
the
left;
the
lower
horizontal
arrow
is
induced
by
the
natural
morphism
of
monoids
Φ
→
Φ
pf
;
the
upper
horizontal
arrow
is
the
natural
functor
C
→
C
pf
of
Definition
3.1,
(iii).
In
particular,
the
functor
C
pf
→
F
Φ
pf
determines
a
pre-Frobenioid
structure
on
C
pf
.
(ii)
An
arrow
of
C
pf
is
a(n)
morphism
of
Frobenius
type
(respectively,
pre-step;
base-isomorphism;
base-identity
endomorphism;
isomorphism;
pull-back
morphism;
isometry;
co-angular
morphism;
LB-invertible
mor-
phism;
morphism
of
a
given
Frobenius
degree)
if
and
only
if
a
cofinal
collection
of
the
system
of
arrows
of
C
that
determine
this
arrow
of
C
pf
[cf.
Definition
3.1,
(ii)]
is
so.
(iii)
The
category
C
pf
,
equipped
with
the
functor
C
pf
→
F
Φ
pf
of
the
diagram
of
(i),
is
a
Frobenioid
of
perfect
and
isotropic
type.
Moreover,
there
is
a
natural
∼
equivalence
of
categories
C
pf
→
(C
pf
)
pf
.
Proof.
In
light
of
our
assumption
that
the
Frobenioid
C
is
of
Frobenius-isotropic
type,
assertions
(i),
(ii),
(iii)
follow
immediately
from
the
definitions;
Proposition
1.10,
(i)
[cf.
also
Proposition
2.1,
(iii)].
Proposition
3.3.
(Base-identity
Pre-steps
and
Units)
(i)
Write
pl-bk
pl-bk
End(C
A
→
C)
bs-iso
⊆
End(C
A
→
C)
pl-bk
[where
C
pl-bk
is
as
in
Definition
1.3,
(i),
(c);
C
A
→
C
is
the
natural
functor]
for
the
submonoid
consisting
of
those
natural
transformations
such
that
the
various
endomorphisms
of
objects
of
C
that
occur
in
the
natural
transformation
are
all
base-
isomorphisms.
Then
if
D
is
Frobenius-slim,
then
the
image
of
1
∈
Z
≥0
⊆
F
under
any
homomorphism
of
monoids
pl-bk
→
C)
bs-iso
F
→
End(C
A
pl-bk
determines
an
element
of
End(C
A
→
C)
bs-iso
with
the
property
that
the
various
endomorphisms
of
objects
of
C
that
occur
in
the
natural
transformation
determined
by
this
element
are
all
base-identity
pre-steps
[i.e.,
lie
in
“O
(−)”].
Conversely,
if
C
is
of
Frobenius-normalized
type,
and
A
is
Frobenius-trivial,
then
every
base-identity
pre-step
endomorphism
of
A
arises
as
the
endomorphism
of
A
induced
pl-bk
by
the
image
of
1
∈
Z
≥0
⊆
F
via
a
homomorphism
of
monoids
F
→
End(C
A
→
bs-iso
.
C)
(ii)
Two
co-objective
morphisms
α
1
,
α
2
:
A
→
B
of
C
istr
are
unit-equivalent
if
and
only
if
they
map
to
the
same
morphism
of
F
Φ
,
i.e.,
if
and
only
if
the
following
three
conditions
are
satisfied:
(a)
deg
Fr
(α
1
)
=
deg
Fr
(α
2
);
(b)
Div(α
1
)
=
Div(α
2
);
(c)
Base(α
1
)
=
Base(α
2
).
(iii)
There
is
a
natural
functor
C
istr
→
C
un-tr
60
SHINICHI
MOCHIZUKI
which
is
full
and
essentially
surjective;
moreover,
this
functor
is
an
equivalence
of
categories
if
and
only
if
C
istr
is
of
unit-trivial
type.
(iv)
The
functor
C
istr
→
F
Φ
factors
naturally
through
C
un-tr
,
hence
determines
a
functor
C
un-tr
→
F
Φ
which
is
faithful
and
essentially
surjective;
moreover,
this
functor
determines
a
natural
structure
of
Frobenioid
on
C
un-tr
,
with
respect
to
which
C
un-tr
is
of
isotropic
and
unit-trivial
type.
Finally,
an
arrow
of
C
un-tr
is
a(n)
morphism
of
Frobenius
type
(respectively,
pre-step;
base-isomorphism;
isomorphism;
pull-back
morphism;
isometry;
co-angular
morphism;
LB-invertible
mor-
phism;
morphism
of
a
given
Frobenius
degree)
if
and
only
if
it
arises
from
such
an
arrow
of
C
istr
.
(v)
The
functor
C
→
F
Φ
is
an
equivalence
of
categories
if
and
only
if
the
Frobenioid
C
is
of
Aut-ample,
unit-trivial,
and
base-trivial
type.
Proof.
First,
we
consider
assertion
(i).
Note
that
since
the
composite
of
the
functor
pl-bk
C
A
→
C
with
the
natural
projection
functor
C
→
D
factors
as
the
composite
of
the
equivalence
of
categories
[involving
pull-back
morphisms]
of
Definition
1.3,
(i),
∼
def
pl-bk
(c),
C
A
→
D
A
D
[where
A
D
=
Base(A)]
with
the
natural
functor
D
A
D
→
D,
it
pl-bk
→
C)
bs-iso
determines
follows
that
any
homomorphism
of
monoids
F
→
End(C
A
a
homomorphism
of
monoids
F
→
Aut(D
A
D
→
D)
—
which,
if
D
is
Frobenius-slim
[cf.
Definition
3.1,
(i)],
necessarily
factors
through
the
natural
surjection
F
N
≥1
—
together
with
a
homomorphism
of
monoids
F
→
N
≥1
obtained
by
considering
the
Frobenius
degree
of
the
induced
endomorphism
of
A
—
which
[in
light
of
the
fact
that
the
monoid
N
≥1
is
commutative,
together
with
the
structure
of
F
—
cf.
Definition
1.1,
(iii)]
also
necessarily
factors
through
the
natural
surjection
F
N
≥1
.
Thus,
we
conclude
that
if
D
is
Frobenius-slim,
then
the
image
pl-bk
→
of
1
∈
Z
≥0
⊆
F
under
the
given
homomorphism
of
monoids
F
→
End(C
A
pl-bk
bs-iso
bs-iso
determines
an
element
of
End(C
A
→
C)
with
the
property
that
the
C)
various
endomorphisms
of
objects
of
C
that
occur
in
the
natural
transformation
determined
by
this
element
are
all
base-identity
pre-steps,
as
desired.
The
“converse
assertion”
[when
C
is
of
Frobenius-normalized
type,
and
A
is
Frobenius-trivial]
may
be
verified
by
choosing
a
homomorphism
of
monoids
N
≥1
→
End
C
(A)
THE
GEOMETRY
OF
FROBENIOIDS
I
61
as
in
the
definition
of
the
term
“Frobenius-trivial”
[cf.
the
homomorphism
“ζ”
of
Definition
1.2,
(iv)],
which,
together
with
the
homomorphism
of
monoids
Z
≥0
→
End
C
(A)
that
maps
1
∈
Z
≥0
to
a
given
base-identity
pre-step
endomorphism
of
A,
yields
[cf.
our
assumption
that
C
is
of
Frobenius-normalized
type!]
a
homomorphism
of
monoids
F
→
End
C
(A)
—
which,
by
applying
Proposition
1.11,
(iii),
lifts
to
a
homomorphism
of
monoids
pl-bk
→
C)
bs-iso
,
as
desired.
This
completes
the
proof
of
assertion
(i).
F
→
End(C
A
Next,
we
consider
assertion
(ii).
Since
assertion
(ii)
clearly
only
concerns
the
Frobenioid
C
istr
[cf.
Proposition
1.9,
(v)],
we
may
replace
C
by
C
istr
and
assume
for
the
remainder
of
the
proof
of
assertion
(ii)
that
C
is
of
isotropic
type.
Now
the
necessity
of
the
three
conditions
(a),
(b),
(c)
follows
immediately
[cf.
Remark
1.1.1]
from
the
fact
that
endomorphisms
of
“O
×
”
are
LB-invertible
base-identity
linear
endomorphisms.
To
show
the
sufficiency
of
these
three
conditions,
we
apply
the
factorization
of
Definition
1.3,
(iv),
(a)
[cf.
also
Proposition
1.4,
(i)],
the
essential
uniqueness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
[cf.
Defi-
nition
1.3,
(ii)],
and
the
equivalence
of
categories
[involving
pull-back
morphisms]
of
Definition
1.3,
(i),
(c),
to
α
1
,
α
2
.
Then
conditions
(a),
(c)
imply
that
there
exist
morphisms
γ
:
A
→
C;
β
1
,
β
2
:
C
→
D;
δ
:
D
→
B,
where
γ
is
a
mor-
phism
of
Frobenius
type,
β
1
and
β
2
are
base-equivalent
co-angular
pre-steps,
and
δ
:
D
→
B
is
a
pull-back
morphism
such
that
α
1
=
δ
◦
β
1
◦
γ,
α
2
=
δ
◦
β
2
◦
γ.
Since
δ,
γ
are
LB-invertible,
it
thus
follows
from
condition
(b)
[cf.
also
Remark
1.1.1]
that
Div(β
1
)
=
Div(β
2
),
hence
[by
Definition
1.3,
(vi)]
that
β
2
=
ζ
◦
β
1
,
for
some
ζ
∈
O
×
(D).
Since
α
1
=
δ
◦
(β
1
◦
γ),
α
2
=
δ
◦
ζ
◦
(β
1
◦
γ),
we
thus
conclude
that
O
×
α
1
≈
α
2
,
as
desired.
This
completes
the
proof
of
assertion
(ii).
Now
assertion
(iii)
is
immediate
from
the
definitions.
In
light
of
assertions
(ii),
(iii),
assertion
(iv)
is
immediate
from
the
definitions.
As
for
assertion
(v),
the
necessity
of
the
condition
in
the
statement
of
assertion
(v)
follows
immediately
from
Proposition
1.5,
(i),
(ii).
To
verify
the
sufficiency
of
this
condition,
let
us
first
observe
that
if
C
is
of
unit-trivial
and
base-trivial
type,
then
[by
the
existence
of
isotropic
hulls
in
C
—
cf.
Definition
1.3,
(vii),
(a)]
it
follows
that
C
is
also
of
isotropic
type,
hence
that
we
have
a
natural
equivalence
of
categories
∼
C
→
C
un-tr
[cf.
assertion
(iii)].
Thus,
by
assertion
(iv),
it
follows
that
the
natural
functor
C
→
F
Φ
is
faithful
and
essentially
surjective.
Since
C
is
of
base-trivial
and
Aut-ample
type,
it
follows
from
the
factorization
of
Definition
1.3,
(iv),
(a)
[cf.
also
the
existence
and
uniqueness
of
morphisms
of
Frobenius
type
of
a
given
Frobenius
degree
asserted
in
Definition
1.3,
(ii);
the
equivalence
of
categories
involving
pull-
back
morphisms
of
Definition
1.3,
(i),
(c)],
that
to
show
that
C
→
F
Φ
is
full,
it
suffices
to
show
that
it
is
surjective
on
base-identity
pre-step
endomorphisms
[i.e.,
on
“O
(−)”];
but,
by
our
assumption
that
C
is
of
base-trivial
and
Aut-ample
type,
this
follows
immediately
from
the
first
equivalence
of
categories
of
Definition
1.3,
(iii),
(d).
This
completes
the
proof
of
assertion
(v).
62
SHINICHI
MOCHIZUKI
Theorem
3.4.
(Category-theoreticity
of
the
Base
and
Frobenius
De-
gree)
For
i
=
1,
2,
let
Φ
i
be
a
divisorial
monoid
on
a
connected,
totally
epimor-
phic
category
D
i
;
C
i
→
F
Φ
i
a
Frobenioid;
∼
Ψ
:
C
1
→
C
2
an
equivalence
of
categories.
Then:
(i)
Suppose
that
C
1
,
C
2
are
of
quasi-isotropic
type.
Then
Ψ
preserves
the
isotropic
objects,
isotropic
hulls,
and
isometric
pre-steps.
Moreover,
there
exists
a
1-unique
functor
Ψ
istr
:
C
1
istr
→
C
2
istr
that
fits
into
a
1-commutative
diagram
C
1
⏐
⏐
C
1
istr
Ψ
−→
C
2
⏐
⏐
Ψ
istr
−→
C
2
istr
[where
the
vertical
arrows
are
the
natural
“isotropification
functors”
of
Proposition
1.9,
(v);
the
horizontal
arrows
are
equivalences
of
categories].
Finally,
if
D
1
,
D
2
are
slim,
and
C
1
,
C
2
are
of
Frobenius-normalized
type,
then
each
of
the
composite
functors
of
this
diagram
is
rigid.
(ii)
Suppose
that
C
1
,
C
2
are
of
quasi-isotropic
type,
and
that
D
1
,
D
2
are
of
FSMFF-type.
Then
Ψ
preserves
pre-steps,
co-angular
pre-steps,
and
group-
like
objects.
(iii)
Suppose
that:
(a)
C
1
,
C
2
are
of
standard
type;
(b)
if
C
1
,
C
2
are
of
group-
like
type,
then
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
base-isomorphisms.
Then
Ψ
preserves
morphisms
of
Frobenius
type,
linear
morphisms,
base-
isomorphisms,
co-angular
morphisms,
pull-back
morphisms,
isometries,
and
LB-invertible
morphisms.
Moreover,
there
exists
an
automorphism
of
monoids
∼
Ψ
N
≥1
:
N
≥1
→
N
≥1
such
that
Ψ
maps
morphisms
of
Frobenius
degree
d
to
morphisms
of
Frobenius
degree
Ψ
N
≥1
(d);
if
C
1
,
C
2
admit
a
non-group-like
object,
then
Ψ
N
≥1
is
the
identity
pf
pf
automorphism.
Also,
there
exists
a
1-unique
functor
Ψ
:
C
1
→
C
2
pf
that
fits
into
a
1-commutative
diagram
Ψ
C
1
−→
C
2
⏐
⏐
⏐
⏐
C
1
pf
Ψ
pf
−→
C
2
pf
[where
the
vertical
arrows
are
the
natural
functors
of
Proposition
3.2,
(i);
the
hori-
zontal
arrows
are
equivalences
of
categories].
Finally,
if
D
1
,
D
2
are
slim,
then
each
of
the
composite
functors
of
this
diagram
is
rigid.
(iv)
Suppose
that:
(a)
C
1
,
C
2
are
of
standard
type;
(b)
if
C
1
,
C
2
are
of
group-
like
type,
then
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
base-isomorphisms;
THE
GEOMETRY
OF
FROBENIOIDS
I
63
(c)
D
1
,
D
2
are
Frobenius-slim.
Then
Ψ
preserves
the
submonoids
“O
(−)”,
“O
×
(−)”;
Ψ
N
≥1
is
the
identity
automorphism.
Moreover,
there
exists
a
1-unique
functor
Ψ
un-tr
:
C
1
un-tr
→
C
2
un-tr
that
fits
into
a
1-commutative
diagram
C
1
istr
⏐
⏐
C
1
un-tr
Ψ
istr
−→
Ψ
un-tr
−→
C
2
istr
⏐
⏐
C
2
un-tr
[where
the
vertical
arrows
are
the
natural
functors
of
Proposition
3.3,
(iii);
the
horizontal
arrows
are
equivalences
of
categories].
Finally,
if
D
1
,
D
2
are
slim,
then
each
of
the
composite
functors
of
this
diagram
is
rigid.
(v)
Suppose
that:
(a)
C
1
,
C
2
are
of
standard
type;
(b)
if
C
1
,
C
2
are
of
group-
like
type,
then
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
base-isomorphisms;
(c)
D
1
,
D
2
are
slim.
Then
Ψ
preserves
the
base-identity
endomorphisms
and
base-equivalent
pairs
of
co-objective
morphisms.
Moreover,
there
exists
a
1-
unique
functor
Ψ
Base
:
D
1
→
D
2
that
fits
into
a
1-commutative
diagram
C
1
⏐
⏐
D
1
Ψ
−→
Ψ
Base
−→
C
2
⏐
⏐
D
2
[where
the
vertical
arrows
are
the
natural
projection
functors;
the
horizontal
ar-
rows
are
equivalences
of
categories].
Finally,
each
of
the
composite
functors
of
this
diagram
is
rigid.
Proof.
First,
we
consider
assertion
(i).
Since
iso-subanchors
are
manifestly
pre-
served
by
any
equivalence
of
categories,
it
follows
from
our
assumption
that
C
1
,
C
2
are
of
quasi-isotropic
type
that
Ψ
preserves
isotropic
objects.
Now,
with
the
excep-
tion
of
the
final
statement
concerning
the
rigidity
of
the
composite
functors,
the
remainder
of
assertion
(i)
follows
formally
from
[the
definitions
and]
Proposition
1.9,
(v),
(vi),
(vii).
The
final
statement
concerning
the
rigidity
of
the
composite
functors
may
be
verified
as
follows:
By
Proposition
1.13,
(ii),
it
suffices
to
show,
for
each
A
∈
Ob(C
istr
)
that
the
automorphism
α
∈
O
×
(A)
induced
by
an
automor-
phism
∈
Aut(C
1
→
C
1
istr
)
is
trivial.
But,
by
Definition
1.3,
(i),
(a),
(b);
(iii),
(c),
it
suffices
to
show
this
when
A
is
Frobenius-trivial,
in
which
case
the
triviality
of
α
follows
from
the
functoriality
of
α
with
respect
to
base-identity
endomorphisms
of
A
of
arbitrary
of
Frobenius
degree
[which
implies,
since
C
1
,
C
2
are
of
Frobenius-
normalized
type,
that
α
d
=
α,
for
all
d
∈
N
≥1
,
hence
that
α
is
trivial,
as
desired].
Next,
we
consider
assertion
(ii).
By
assertion
(i)
[cf.
also
Proposition
1.9,
(v)],
and
the
characterization
of
co-angular
pre-steps
given
in
Proposition
1.7,
(iv),
we
reduce
immediately
to
the
case
where
C
1
,
C
2
are
of
isotropic
type.
Then
[since
any
equivalence
of
categories
manifestly
preserves
FSM-morphisms
and
irreducible
mor-
phisms]
the
fact
that
Ψ
preserves
pre-steps
follows
formally
from
Proposition
1.14,
(ii),
(iii).
Since
Ψ
preserves
pre-steps,
it
thus
follows
from
Proposition
1.8,
(iii)
[cf.
64
SHINICHI
MOCHIZUKI
also
Proposition
1.4,
(i)],
that
Ψ
preserves
group-like
objects.
This
completes
the
proof
of
assertions
(i),
(ii).
Next,
we
consider
assertion
(iii).
First,
I
claim
that
to
verify
assertion
(iii),
it
suffices
to
prove
that,
for
each
prime
p
1
∈
Primes,
there
exists
a
prime
p
2
∈
Primes,
which
is
equal
to
p
1
if
C
1
,
C
2
are
not
of
group-like
type,
such
that
Ψ
istr
maps
p
1
-
Frobenius
morphisms
to
p
2
-Frobenius
morphisms.
Indeed,
the
assignment
p
1
→
p
2
determines
a
homomorphism
of
monoids
Ψ
N
≥1
:
N
≥1
→
N
≥1
which
[by
considering
a
quasi-inverse
to
Ψ]
is
easily
seen
to
be
an
automorphism.
Moreover,
by
Proposition
1.10,
(v),
the
condition
of
the
claim
implies
that
Ψ
istr
preserves
morphisms
of
Frobenius
type,
hence
also
linear
morphisms
[by
Proposition
1.7,
(iii)],
and
maps
morphisms
of
Frobenius
degree
d
to
morphisms
of
Frobenius
degree
Ψ
N
≥1
(d)
[i.e.,
since
arbitrary
morphisms
may
be
written
as
composites
of
prime-Frobenius
morphisms
and
linear
morphisms
—
cf.
Remark
1.1.1;
Definition
1.3,
(iv),
(a);
Proposition
1.10,
(v)].
Since
the
isotropification
functor
preserves
Frobenius
degrees,
this
implies
that
Ψ
maps
morphisms
of
Frobenius
degree
d
to
morphisms
of
Frobenius
degree
Ψ
N
≥1
(d),
hence
that
Ψ
preserves
linear
morphisms
and
morphisms
of
Frobenius
type
[by
Proposition
1.7,
(iii)].
Moreover,
by
assertions
(i),
(ii),
Ψ
preserves
isometric
pre-steps
and
pre-steps,
hence
base-isomorphisms
[i.e.,
composites
of
pre-steps
and
morphisms
of
Frobenius
type
—
cf.
Proposition
1.7,
(ii)],
pull-back
morphisms
[cf.
Proposition
1.7,
(ii)],
isometries
[i.e.,
morphisms
that
map
via
the
isotropification
functor
to
composites
of
a
morphism
of
Frobenius
type
and
a
pull-back
morphism
—
cf.
Propositions
1.4,
(i),
(v);
1.9,
(v)],
co-angular
morphisms
[cf.
Definition
1.2,
(iii);
assertion
(i)
for
isometric
pre-steps],
and
LB-
invertible
morphisms.
Now
it
follows
immediately
from
the
definition
of
C
pf
[cf.
Definition
3.1,
(iii)]
that
we
obtain
a
1-unique
1-commutative
diagram
as
in
the
statement
of
assertion
(iii).
Finally,
to
verify
the
asserted
rigidity
of
composite
functors,
it
suffices
[cf.
the
argument
applied
in
the
proof
of
assertion
(i)]
to
apply
Proposition
1.13,
(ii),
and
to
consider
the
functoriality
of
the
automorphisms
in
question
with
respect
to
base-identity
endomorphisms
of
Frobenius-trivial
objects
of
arbitrary
Frobenius
degree.
This
completes
the
proof
of
the
claim.
Thus,
to
complete
the
proof
of
assertion
(iii),
we
may
assume
[for
the
remainder
of
the
proof
of
assertion
(iii)]
that
C
1
,
C
2
are
of
isotropic
type
[cf.
assertion
(i)].
Then
it
suffices
to
prove
that,
for
each
prime
p
1
∈
Primes,
there
exists
a
prime
p
2
∈
Primes,
which
is
equal
to
p
1
if
C
1
,
C
2
are
not
of
group-like
type,
such
that
Ψ
maps
p
1
-
Frobenius
morphisms
to
p
2
-Frobenius
morphisms.
Let
us
call
an
object
A
1
∈
Ob(C
1
)
(p
1
,
p
2
)-admissible
if
Ψ
maps
every
p
1
-Frobenius
morphism
with
domain
A
1
to
a
p
2
-Frobenius
morphism
of
C
2
.
Now
let
us
consider
the
following
assertions:
(F1)
For
each
prime
p
1
∈
Primes,
there
exist
a
prime
p
2
∈
Primes
and
a
(p
1
,
p
2
)-admissible
object
of
C
1
.
(F2)
For
every
pair
of
primes
p
1
,
p
2
∈
Primes
and
every
morphism
ζ
1
:
A
1
→
B
1
in
C
1
,
A
1
is
(p
1
,
p
2
)-admissible
if
and
only
if
B
1
is.
THE
GEOMETRY
OF
FROBENIOIDS
I
65
(F3)
If
C
1
,
C
2
are
not
of
group-like
type,
then
for
each
prime
p
∈
Primes,
there
exist
a
(p,
p)-admissible
object
of
C
1
.
Observe,
moreover,
that
since
C
1
is
connected,
to
complete
the
proof
of
assertion
(iii),
it
suffices
to
prove
(F1),
(F2),
(F3).
First,
we
consider
assertion
(F1).
Let
us
first
consider
the
case
where
C
1
,
C
2
are
of
group-like
type.
Then
all
pre-steps
of
C
1
,
C
2
are
isomorphisms;
Ψ
preserves
base-isomorphisms.
Thus,
for
any
A
1
∈
Ob(C
1
),
the
prime-Frobenius
morphisms
with
domain
A
1
are
precisely
the
irreducible
base-isomorphisms
with
domain
A
1
[cf.
Proposition
1.14,
(i)].
In
particular,
Ψ
preserves
the
prime-Frobenius
morphisms;
hence,
we
conclude
that
assertion
(F1)
holds.
Next,
let
us
consider
the
case
where
C
1
,
C
2
are
not
of
group-like
type.
Then
if
A
1
is
non-group-like,
then
[cf.
Definition
1.3,
(i),
(a);
Proposition
1.8,
(iii)],
there
exists
a
base-isomorphic
[i.e.,
to
A
1
],
hence
non-group-like,
Frobenius-trivial
object
of
C
1
.
Thus,
we
may
assume
without
loss
of
generality
that
A
1
is
Frobenius-trivial.
Then
for
any
p
1
∈
Primes,
there
exists
a
base-identity
[hence
Div-identity]
p
1
-Frobenius
endomorphism
φ
1
of
A
1
.
Since
[by
assertion
(ii)]
Ψ
preserves
pre-steps,
it
thus
follows
formally
from
the
characteriza-
tion
of
“Div-identity
prime-Frobenius
endomorphisms”
given
in
Proposition
1.14,
def
(v),
that
Ψ
maps
φ
1
to
a
prime-Frobenius
endomorphism
of
A
2
=
Ψ(A
1
).
This
completes
the
proof
of
assertion
(F1).
Next,
we
consider
assertion
(F2).
First,
observe
that
if
the
morphism
ζ
1
:
A
1
→
B
1
is
a
pre-step,
then
[since,
by
assertion
(ii),
Ψ
preserves
pre-steps]
it
follows
by
applying
Proposition
1.14,
(iv),
to
commutative
diagrams
such
as
the
one
given
in
Proposition
1.10,
(i),
that
assertion
(F2)
holds.
Thus,
by
Definition
1.3,
(i),
(a),
(b),
(c),
we
may
assume
without
loss
of
generality
that
B
1
is
Frobenius-trivial,
and
that
ζ
1
is
a
pull-back
morphism.
Now,
by
applying
Proposition
1.11,
(iii),
it
follows
that
for
every
p
1
∈
Primes,
there
exist
base-identity
p
1
-Frobenius
endomorphisms
φ
1
∈
End
C
(A
1
),
ψ
1
∈
End
C
(B
1
)
such
that
ψ
1
◦
ζ
1
=
ζ
1
◦
φ
1
.
In
particular,
if
we
def
def
def
write
φ
2
=
Ψ(φ
1
),
ψ
2
=
Ψ(ψ
1
),
ζ
2
=
Ψ(ζ
1
),
then
ψ
2
◦
ζ
2
=
ζ
2
◦
φ
2
,
and
φ
2
,
ψ
2
are
irreducible.
Thus,
by
Proposition
1.14,
(iv),
we
obtain
that
φ
2
is
a
p
2
-Frobenius
morphism
if
and
only
if
ψ
2
is.
This
completes
the
proof
of
assertion
(F2).
Finally,
we
consider
assertion
(F3).
Let
A
1
∈
Ob(C
1
)
be
a
non-group-like,
Frobenius-trivial
object
[cf.
the
proof
of
assertion
(F1)].
By
assertions
(F1),
(F2),
it
follows
already
that
Ψ
preserves
prime-Frobenius
morphisms.
Thus,
to
complete
the
proof
of
assertion
(F3),
[since
the
Frobenius
degree
of
a
prime-Frobenius
mor-
phism
is
always
a
prime
number]
it
suffices
to
show
that
if
ζ
1
,
θ
1
∈
End
C
1
(A
1
)
are
prime-Frobenius
base-identity
endomorphisms
such
that
deg
Fr
(ζ
1
)
<
deg
Fr
(θ
1
),
def
def
then
deg
Fr
(ζ
2
)
<
deg
Fr
(θ
2
)
[where
ζ
2
=
Ψ(ζ
1
),
θ
2
=
Ψ(θ
1
)].
But,
by
the
first
equivalence
of
categories
of
Definition
1.3,
(iii),
(d)
[cf.
also
Proposition
1.10,
(i)],
the
condition
“deg
Fr
(ζ
1
)
<
deg
Fr
(θ
1
)”
is
equivalent
to
the
following
condition:
If
we
write
β
ζ
1
(respectively,
β
θ
1
)
for
the
step
“β
◦
α”
of
Proposition
1.14,
(v),
obtained
when
one
takes
“φ”
of
loc.
cit.
to
be
ζ
1
(respectively,
θ
1
)
[and
“α”
of
loc.
cit.
to
be
some
fixed
step],
then
β
θ
1
=
γ
1
◦
β
ζ
1
,
for
some
step
γ
1
.
66
SHINICHI
MOCHIZUKI
Thus,
if
we
write
β
ζ
2
(respectively,
β
θ
2
)
for
the
step
“β
◦
α”
of
Proposition
1.14,
(v),
obtained
when
one
takes
“φ”
of
loc.
cit.
to
be
ζ
2
(respectively,
θ
2
)
[and
“α”
of
loc.
cit.
to
be
some
fixed
step],
then
β
θ
2
=
γ
2
◦
β
ζ
2
,
for
some
step
γ
2
[since,
by
assertion
(ii),
we
already
know
that
Ψ
preserves
pre-steps],
which
[again
by
the
first
equivalence
of
categories
of
Definition
1.3,
(iii),
(d);
Proposition
1.10,
(i)]
implies
that
deg
Fr
(ζ
2
)
<
deg
Fr
(θ
2
),
as
desired.
This
completes
the
proof
of
assertion
(F3),
hence
also
the
proof
of
assertion
(iii).
Next,
let
us
observe
that
by
assertion
(i)
[cf.
also
Proposition
1.9,
(v)],
it
suffices
to
verify
assertions
(iv),
(v),
under
the
further
assumption
that
C
1
,
C
2
are
of
isotropic
type;
thus,
we
assume
for
the
remainder
of
the
proof
of
Theorem
3.4
that
C
1
,
C
2
are
of
isotropic
type.
Also,
to
simplify
notation
[for
the
remainder
of
the
proof
of
Theorem
3.4],
let
us
write
def
P
i
=
C
i
pl-bk
[cf.
Definition
1.3,
(i),
(c)],
for
i
=
1,
2.
Next,
let
us
consider
assertion
(iv).
Now,
for
i
=
1,
2,
it
follows
formally
[in
light
of
our
assumption
that
D
i
is
Frobenius-slim]
from
Proposition
3.3,
(i)
[cf.
also
Definition
1.3,
(i),
(a),
(b);
(iii),
(c)],
that
if
C
∈
Ob(C
i
),
then
the
endomorphisms
of
O
(C)
are
precisely
the
endomorphisms
γ
∈
End
C
i
(C)
such
that
the
following
condition
is
satisfied:
There
exist
pre-steps
φ
:
A
→
B,
ψ
:
A
→
C
and
endomorphisms
α
∈
End
C
i
(A),
β
∈
End
C
i
(B)
such
that
β
◦
φ
=
φ
◦
α,
γ
◦
ψ
=
ψ
◦
α,
and,
moreover,
α
arises
as
the
endomorphism
of
A
induced
by
the
image
of
1
∈
Z
≥0
⊆
F
via
a
homomorphism
of
monoids
F
→
End((P
i
)
A
→
C
i
)
bs-iso
.
By
assertions
(ii),
(iii),
it
follows
that
Ψ
preserves
pre-steps,
base-isomorphisms,
and
pull-back
morphisms,
hence
that
Ψ
preserves
endomorphisms
satisfying
the
above
condition.
Thus,
we
conclude
that
Ψ
preserves
the
submonoids
“O
(−)”,
“O
×
(−)”,
as
desired.
The
existence
of
a
a
1-unique
functor
Ψ
un-tr
:
C
1
un-tr
→
C
2
un-tr
that
fits
into
a
1-commutative
diagram
as
in
the
statement
of
assertion
(iv)
then
follows
formally
from
the
definition
of
“C
1
un-tr
”,
“C
2
un-tr
”;
since
“C
1
un-tr
”,
“C
2
un-tr
”
are
of
unit-trivial
type,
the
asserted
rigidity
follows
formally
from
Proposition
1.13,
(ii).
Thus,
to
complete
the
proof
of
assertion
(iv),
it
suffices
to
show
that
Ψ
N
≥1
is
the
identity
automorphism.
If
C
1
,
C
2
are
not
of
group-like
type,
then
this
already
follows
formally
from
assertion
(iii).
Thus,
let
us
assume
for
the
remainder
of
the
proof
of
assertion
(iv)
that
C
1
,
C
2
are
of
group-like
type.
Observe
that
there
exists
def
an
object
A
1
∈
Ob(C
1
)
such
that
A
2
=
Ψ(A
1
)
is
Frobenius-compact
[cf.
Definition
3.1;
the
fact
that
Ψ
is
an
equivalence
of
categories].
By
Proposition
1.10,
(vi),
A
1
,
A
2
are
Frobenius-trivial.
Let
φ
1
∈
End
C
1
(A
1
)
be
a
base-identity
prime-Frobenius
def
endomorphism.
By
assertion
(iii),
φ
2
=
Ψ(φ
1
)
is
also
a
prime-Frobenius
morphism.
Write
φ
2
=
α
◦
ψ
2
,
where
ψ
2
is
a
base-identity
prime-Frobenius
endomorphism
of
THE
GEOMETRY
OF
FROBENIOIDS
I
67
A
2
,
and
α
∈
Aut
C
2
(A
2
)
[cf.
Definition
1.3,
(ii)].
Now
since
C
1
,
C
2
are
of
Frobenius-
normalized
type
[cf.
Definition
3.1,
(i),
(c)],
it
follows
that
for
every
u
1
∈
O
×
(A
1
),
def
u
p
1
1
◦
φ
1
=
φ
1
◦
u
1
[where
p
1
=
deg
Fr
(φ
1
)].
Thus,
for
u
2
∈
O
×
(A
2
),
we
obtain
u
p
2
1
◦
φ
2
=
φ
2
◦
u
2
=
α
◦
ψ
2
◦
u
2
=
α
◦
u
p
2
2
◦
ψ
2
=
α
◦
u
p
2
2
◦
α
−1
◦
α
◦
ψ
2
=
α
◦
u
p
2
2
◦
α
−1
◦
φ
2
def
[where
p
2
=
deg
Fr
(φ
2
)],
hence
[by
the
total
epimorphicity
of
C
2
]
u
p
2
1
=
α
◦
u
p
2
2
◦
α
−1
—
i.e.,
α
acts
on
O
×
(A
2
)
pf
by
multiplication
by
p
1
/p
2
.
Since
A
2
is
Frobenius-
compact,
we
thus
conclude
that
p
1
=
p
2
.
This
completes
the
proof
of
assertion
(iv).
Finally,
we
consider
assertion
(v).
Now,
for
i
=
1,
2,
if
A
∈
Ob(C
i
)
=
Ob(P
i
),
def
A
D
=
Base(A)
∈
Ob(D
i
),
then
the
natural
projection
functor
C
i
→
D
i
determines
a
natural
equivalence
of
categories
∼
(P
i
)
A
→
(D
i
)
A
D
def
[cf.
Definition
1.3,
(i),
(c)].
Moreover,
if
A
∈
Ob(C
i
)
=
Ob(P
i
),
A
D
=
Base(A
)
∈
Ob(D
i
),
then
any
arrow
A
→
A
determines
a
functor
(P
i
)
A
→
(P
i
)
A
by
sending
an
object
φ
:
C
→
A
of
(P
i
)
A
to
the
object
C
→
A
of
(P
i
)
A
which
is
the
pull-back
morphism
of
C
i
that
appears
in
the
factorization
of
the
composite
C
→
A
→
A
given
in
Definition
1.3,
(iv),
(a).
Moreover,
one
verifies
immediately
that
this
functor
fits
into
a
natural
1-commutative
diagram
(P
i
)
A
⏐
⏐
−→
(D
i
)
A
D
−→
(D
i
)
A
D
(P
i
)
A
⏐
⏐
[where
the
upper
horizontal
arrow
is
the
functor
just
defined;
the
vertical
arrows
are
the
equivalences
that
arise
from
the
natural
projection
functor
C
i
→
D
i
;
the
lower
horizontal
arrow
is
the
natural
functor
[cf.
§0]
determined
by
the
arrow
A
D
→
A
D
obtained
by
projecting
the
given
arrow
A
→
A
to
D
i
].
Next,
observe
that
since
the
category
D
i
,
hence
also
the
categories
(D
i
)
A
D
,
(P
i
)
A
,
are
slim,
it
follows
that
the
collection
of
categories
“(P
i
)
A
”
[where
i
is
fixed;
A
ranges
over
the
objects
of
C
i
]
and
functors
“(P
i
)
A
→
(P
i
)
A
”
[arising
from
arrows
A
→
A
of
C
i
]
determine
a
2-slim
[cf.
Definition
A.1,
(i)]
2-category
of
1-categories,
whose
“coarsification”
[cf.
Definition
A.1,
(ii)]
we
denote
by
Q
i
,
together
with
a
natural
functor
C
i
→
Q
i
68
SHINICHI
MOCHIZUKI
[i.e.,
which
maps
A
→
(P
i
)
A
,
(A
→
A)
→
{(P
i
)
A
→
(P
i
)
A
}].
Similarly,
the
collection
of
categories
“(D
i
)
A
D
”
[where
i
is
fixed;
A
D
ranges
over
the
objects
of
D
i
]
and
functors
“(D
i
)
A
D
→
(D
i
)
A
D
”
[arising
from
arrows
A
D
→
A
D
of
D
i
]
determine
a
2-category
of
1-categories,
whose
coarsification
we
denote
by
E
i
,
together
with
a
natural
functor
D
i
→
E
i
—
which,
in
fact,
may
be
identified
with
the
“slim
exponentiation
functor”
of
Propo-
sition
A.2,
hence,
in
particular,
is
an
equivalence
of
categories.
Thus,
since
the
nat-
ural
projection
functor
C
i
→
D
i
is
essentially
surjective
[cf.
Definition
1.3,
(i),
(a)],
it
follows
that
the
natural
projection
functor
C
i
→
D
i
induces
a
faithful,
essentially
surjective
functor
Q
i
→
E
i
∼
which
may
be
composed
with
a
quasi-inverse
to
the
natural
equivalence
D
i
→
E
i
just
discussed
to
obtain
a
faithful,
essentially
surjective
functor
Q
i
→
D
i
[which
is
well-defined
up
to
isomorphism].
def
def
Next,
let
us
observe
that
if
A,
A
∈
Ob(C
i
),
A
D
=
Base(A),
A
D
=
Base(A
),
then
any
morphism
φ
D
:
A
D
→
A
D
may
be
written
in
the
form
φ
D
=
Base(ψ)
◦
Base(γ)
◦
Base(α)
−1
—
where
α
:
B
→
A,
γ
:
B
→
C,
are
pre-steps;
ψ
:
C
→
A
is
a
pull-back
morphism
[cf.
Definition
1.3,
(i),
(b),
(c)].
Since
[by
the
above
discussion]
any
base-
∼
isomorphism
ζ
:
D
→
E
of
C
i
induces
an
equivalence
of
categories
(P
i
)
D
D
→
(P
i
)
E
D
def
def
[where
D,
E
∈
Ob(C
i
),
D
D
=
Base(D),
E
D
=
Base(E)],
it
thus
follows
that
any
collection
of
morphisms
α,
γ,
ψ
as
just
described
determine
a
“new
functor”
(P
i
)
A
D
→
(P
i
)
A
D
[i.e.,
by
inverting
the
equivalence
of
categories
induced
by
α
and
then
composing
with
the
functors
induced
by
γ,
ψ].
Thus,
by
enlarging
the
2-slim
2-category
of
1-categories
considered
above
[i.e.,
whose
coarsification
we
called
Q
i
]
by
considering
these
“new
functors”,
we
obtain
a
[slightly
larger]
2-slim
2-category
of
1-categories,
whose
coarsification
we
denote
by
R
i
.
In
particular,
we
obtain
a
[faithful]
embed-
ding
Q
i
→
R
i
with
the
property
that
the
functor
Q
i
→
D
i
considered
above
admits
a
natural
extension
to
a
functor
R
i
→
D
i
which
[by
the
above
discussion]
is
clearly
an
equivalence
of
categories.
On
the
other
hand,
since
[by
assertions
(ii),
(iii)]
Ψ
preserves
pre-steps,
pull-
back
morphisms,
and
factorizations
as
in
Definition
1.3,
(iv),
(a),
it
follows
that
Ψ
THE
GEOMETRY
OF
FROBENIOIDS
I
69
induces
a
1-commutative
diagram
Ψ
C
1
⏐
⏐
−→
Q
1
⏐
⏐
−→
R
1
−→
R
2
Ψ
Q
C
2
⏐
⏐
Q
2
⏐
⏐
Ψ
R
—
where
the
vertical
functors
are
the
natural
functors
of
the
above
discussion,
and
the
the
horizontal
functors
are
equivalences
of
categories
induced
by
Ψ.
Thus,
∼
by
composing
with
the
natural
equivalences
of
categories
R
i
→
D
i
of
the
above
discussion,
we
obtain
a
1-commutative
diagram
as
in
the
statement
of
assertion
(v),
which
is
clearly
1-unique
[cf.
Definition
1.3,
(i),
(a),
(b),
(c)].
Finally,
the
asserted
rigidity
follows
formally
from
Proposition
1.13,
(i).
This
completes
the
proof
of
assertion
(v).
Remark
3.4.1.
With
regard
to
assumption
(b)
of
Theorem
3.4,
(iii),
(iv),
(v),
we
observe
the
following:
Suppose,
in
the
situation
of
Theorem
3.4,
that
C
1
,
C
2
are
of
group-like
and
quasi-isotropic
type.
Then
if
Ψ
and
some
quasi-inverse
to
Ψ
preserve
Frobenius
degrees,
then
they
also
preserve
base-isomorphisms.
Indeed,
by
Theorem
3.4,
(i),
we
may
assume,
without
loss
of
generality,
that
C
1
,
C
2
are
of
isotropic
type.
Then
if
Ψ
and
some
quasi-inverse
to
Ψ
preserve
Frobenius
degrees,
then
they
preserve
linear
morphisms,
hence
morphisms
of
Frobenius
type
[cf.
Proposition
1.7,
(iii)]
and
base-isomorphisms
[i.e.,
morphisms
of
Frobenius
type,
since
C
1
,
C
2
are
of
group-like
and
isotropic
type
—
cf.
Propositions
1.4,
(i);
1.7,
(ii);
1.8,
(iii)].
One
way
to
understand
the
meaning
of
the
conditions
imposed
in
the
various
portions
of
Theorem
3.4
is
by
considering
examples
in
which
some
of
the
conditions
hold,
but
others
do
not.
Example
3.5.
Base
Categories
with
FSMI-endomorphisms.
Let
D
be
a
one-object
category
whose
unique
object
has
endomorphism
monoid
F;
C
a
one-
object
category
whose
unique
object
has
endomorphism
monoid
F
×
F.
Thus,
the
projection
F
×
F
→
F
to
the
first
factor
determines
a
functor
C
→
D;
C
may
be
identified
with
the
elementary
Frobenioid
determined
by
the
[manifestly
non-
dilating]
monoid
on
D
that
assigns
to
the
unique
object
of
D
the
monoid
Z
≥0
and
to
every
morphism
of
D
the
identity
automorphism
of
Z
≥0
.
In
particular,
C
is
a
Frobenioid
of
Frobenius-normalized
and
isotropic
type,
which
is
not
of
group-like
type
[cf.
Proposition
1.5,
(i),
(ii)].
On
the
other
hand,
one
verifies
immediately
that
every
morphism
of
D
is
an
FSM-morphism,
and
that
the
endomorphism
1
∈
Z
≥0
⊆
F
of
the
unique
object
of
D
is
irreducible.
Thus,
D
admits
an
FSMI-endomorphism,
which
implies
[cf.
§0]
that
D
fails
to
be
of
FSMFF-type.
Moreover,
the
self-
equivalence
of
C
determined
by
the
automorphism
of
monoids
∼
F
×
F
→
F
×
F
70
SHINICHI
MOCHIZUKI
given
by
switching
the
two
factors
clearly
fails
to
preserve
pre-steps
[cf.
Theorem
3.4,
(ii)].
Example
3.6.
Frobenioids
of
Standard
and
Group-like
Type.
Let
def
Z/pZ
G
=
Z
⊕
p∈Primes
[regarded
as
an
abelian
group];
D
a
one-object
category
whose
unique
object
has
endomorphism
monoid
F
G
;
C
a
one-object
category
whose
unique
object
has
endo-
morphism
monoid
F
G
×
F
G
.
Thus,
if
A
∈
Ob(D),
then
each
automorphism
of
an
object
of
D
arising
from
an
element
Aut(D
A
→
D)
is
contained
in
the
subgroup
of
infinitely
divisible
elements
of
G
[cf.
the
proof
of
Proposition
1.13,
(iii)],
hence
is
trivial
—
that
is
to
say,
D
is
slim.
Moreover,
the
projection
F
G
×
F
G
→
F
G
to
the
first
factor
determines
a
functor
C
→
D;
C
may
be
identified
with
the
elementary
Frobenioid
determined
by
the
[manifestly
non-dilating]
monoid
on
D
that
assigns
to
the
unique
object
of
D
the
monoid
G
and
to
every
morphism
of
D
the
identity
auto-
morphism
of
G.
In
particular,
C
is
a
Frobenioid
of
Frobenius-normalized,
isotropic,
and
group-like
type
[cf.
Proposition
1.5,
(i),
(iii)].
One
verifies
immediately
that
every
morphism
of
D
is
either
an
isomorphism
or
a
non-monomorphism
[cf.
the
existence
of
the
torsion
subgroup
p∈Primes
Z/pZ
⊆
G],
and
that
the
irreducible
morphisms
of
D
are
precisely
the
morphisms
that
project
via
the
natural
surjection
F
G
→
N
≥1
to
primes
of
N
≥1
.
Thus,
it
follows
immediately
that
D
is
of
FSM-,
hence
also
of
FSMFF-type.
Moreover,
since
G
pf
∼
=
Q
=
0,
and
the
first
factor
of
F
G
in
the
product
F
G
×
F
G
commutes
with
the
G
[i.e.,
“O
×
(−)”]
of
the
second
factor
of
F
G
,
it
follows
that
the
unique
object
of
C
is
Frobenius-compact.
Thus,
C
is
of
standard
type.
On
the
other
hand,
the
self-equivalence
of
C
determined
by
the
automorphism
of
monoids
∼
F
G
×
F
G
→
F
G
×
F
G
given
by
switching
the
two
factors
clearly
fails
to
preserve
base-isomorphisms
[cf.
Theorem
3.4,
(iii)].
Example
3.7.
Dilating
Monoids.
Let
G,
D
be
as
in
Example
3.6;
Φ
the
monoid
on
D
that
associates
to
the
unique
object
of
D
the
monoid
G
×
Z
≥0
and
to
a
morphism
f
∈
F
G
of
D
that
projects
to
an
element
d
f
∈
N
≥1
the
endomorphism
of
G
×
Z
≥0
that
acts
trivially
on
G
and
by
multiplication
by
d
f
on
Z
≥0
.
Thus,
[as
observed
in
Example
3.6]
D
is
of
FSMFF-type,
but
Φ
clearly
fails
to
be
non-dilating.
def
Write
C
=
F
Φ
.
Thus,
C
is
a
Frobenioid
of
Frobenius-normalized
and
isotropic
type,
which
is
not
of
group-like
type
[cf.
Proposition
1.5,
(i),
(ii)].
Moreover,
C
is
a
one-
object
category
whose
unique
object
has
endomorphism
monoid
M
given
by
the
product
set
Z
≥0
×
(F
G
×
F
G
)
equipped
with
the
following
monoid
structure:
If
a
1
,
a
2
∈
Z
≥0
;
b
1
,
b
2
∈
F
G
×
F
G
,
where
b
1
projects
to
an
element
(n,
m)
∈
N
≥1
×
N
≥1
,
then
(a
1
,
b
1
)
·
(a
2
,
b
2
)
=
(a
1
+
n
·
m
·
a
2
,
b
1
·
b
2
)
THE
GEOMETRY
OF
FROBENIOIDS
I
71
[cf.
the
description
of
elementary
Frobenioids
in
Definition
1.1,
(iii)].
Thus,
by
switching
the
two
factors
of
F
G
,
and
keeping
the
unique
factor
of
Z
≥0
fixed,
we
obtain
an
automorphism
of
the
monoid
M
,
hence
a
self-equivalence
of
C,
that
preserves
pre-steps
[cf.
Theorem
3.4,
(ii)],
but
fails
to
preserve
base-isomorphisms
[cf.
Theorem
3.4,
(iii)].
Example
3.8.
∼
Permutation
of
Primes.
Let
α
:
N
≥1
→
N
≥1
be
an
automor-
def
def
def
phism
of
monoids
of
order
2;
N
=
(N
≥1
)
gp
[so
α
acts
on
N
];
U
=
Q;
V
=
Q;
def
def
W
=
Q;
G
=
U
N
,
where
we
let
n
∈
N
(⊆
Q)
act
on
U
by
n
−1
;
D
the
one-object
category
whose
unique
object
has
endomorphism
monoid
G;
Φ
the
[manifestly
non-
dilating]
monoid
on
D
that
associates
to
the
unique
object
of
D
the
monoid
V
×
W
and
to
a
morphism
g
∈
G
that
projects
to
an
element
n
∈
N
the
automorphism
of
V
×
W
given
by
(α(n),
α(n)
·
n
−1
)
[i.e.,
the
automorphism
that
acts
on
V
by
α(n)
def
and
on
W
by
α(n)
·
n
−1
];
C
=
F
Φ
.
Thus,
C
is
a
Frobenioid
of
Frobenius-normalized,
isotropic,
and
group-like
type
[cf.
Proposition
1.5,
(i),
(iii)];
the
category
D
is
man-
ifestly
of
FSM-,
hence
also
of
FSMFF-type
[cf.
§0].
Since
the
unique
object
of
C
has
“O
×
(−)”
equal
to
V
×
W
,
it
follows
from
our
definition
of
Φ
that
this
object
is
Frobenius-compact.
Thus,
C
is
of
standard
type.
On
the
other
hand,
if
A
∈
Ob(D),
then
Aut(D
A
→
D)
∼
=
G;
since
there
exist
injections
of
monoids
F
→
G,
it
thus
follows
that
D
fails
to
be
Frobenius-slim.
The
monoid
M
of
endomorphisms
of
the
unique
object
of
C
may
be
described
as
the
product
set
U
×
V
×
W
×
N
×
N
≥1
equipped
with
the
following
monoid
structure:
if
u
1
,
u
2
∈
U
;
v
1
,
v
2
∈
V
;
w
1
,
w
2
∈
W
;
n
1
,
n
2
∈
N
;
m
1
,
m
2
∈
N
≥1
,
then
(u
1
,v
1
,
w
1
,
n
1
,
m
1
)
·
(u
2
,
v
2
,
w
2
,
n
2
,
m
2
)
=
−1
(u
1
+
n
−1
1
·
u
2
,
v
1
+
m
1
·
α(n
1
)
·
v
2
,
w
1
+
m
1
·
α(n
1
)
·
n
1
·
w
2
,
n
1
·
n
2
,
m
1
·
m
2
)
[cf.
the
description
of
elementary
Frobenioids
in
Definition
1.1,
(iii)].
In
particular,
a
routine
verification
reveals
that
the
assignment
(u,
v,
w,
n,
m)
→
(v,
u,
w,
α(n)
−1
·
m
−1
,
α(m))
[where
u
∈
U
,
v
∈
V
,
w
∈
W
,
n
∈
N
,
m
∈
N
≥1
]
determines
an
automorphism
of
the
monoid
M
,
hence
a
self-equivalence
of
C,
which
clearly
preserves
base-isomorphisms,
but
fails
to
preserve
“O
×
(−)”
[i.e.,
the
subspace
{0}
×
V
×
W
⊆
U
×
V
×
W
]
or
Frobenius
degrees
[when
α
is
not
equal
to
the
identity]
—
cf.
Theorem
3.4,
(iii),
(iv).
Example
3.9.
def
def
def
def
Non-preservation
of
Units.
Let
N
=
(N
≥1
)
gp
;
U
=
Q;
def
V
=
Q;
W
=
Z
≥0
;
G
=
U
N
,
where
we
let
n
∈
N
(⊆
Q)
act
on
U
by
n
−1
;
D
the
one-object
category
whose
unique
object
has
endomorphism
monoid
G;
Φ
the
[manifestly
non-dilating]
monoid
on
D
that
associates
to
the
unique
object
of
72
SHINICHI
MOCHIZUKI
D
the
monoid
V
×
W
and
to
a
morphism
g
∈
G
that
projects
to
an
element
n
∈
N
the
automorphism
of
V
×
W
given
by
(n,
1)
[i.e.,
the
automorphism
that
acts
on
def
V
by
n
and
on
W
by
1];
C
=
F
Φ
.
Thus,
C
is
a
Frobenioid
of
Frobenius-normalized
and
isotropic
type,
which
is
not
of
group-like
type
[cf.
Proposition
1.5,
(i),
(ii)];
D
is
manifestly
of
FSM-,
hence
also
of
FSMFF-type
[cf.
§0].
Thus,
C
is
of
standard
type.
On
the
other
hand,
[cf.
Example
3.8]
D
fails
to
be
Frobenius-slim.
The
monoid
M
of
endomorphisms
of
the
unique
object
of
C
may
be
described
as
the
product
set
U
×
V
×
W
×
N
×
N
≥1
equipped
with
the
following
monoid
structure:
if
u
1
,
u
2
∈
U
;
v
1
,
v
2
∈
V
;
w
1
,
w
2
∈
W
;
n
1
,
n
2
∈
N
;
m
1
,
m
2
∈
N
≥1
,
then
(u
1
,v
1
,
w
1
,
n
1
,
m
1
)
·
(u
2
,
v
2
,
w
2
,
n
2
,
m
2
)
=
(u
1
+
n
−1
1
·
u
2
,
v
1
+
m
1
·
n
1
·
v
2
,
w
1
+
m
1
·
w
2
,
n
1
·
n
2
,
m
1
·
m
2
)
[cf.
the
description
of
elementary
Frobenioids
in
Definition
1.1,
(iii)].
In
particular,
a
routine
verification
reveals
that
the
assignment
(u,
v,
w,
n,
m)
→
(v,
u,
w,
n
−1
·
m
−1
,
m)
[where
u
∈
U
,
v
∈
V
,
w
∈
W
,
n
∈
N
,
m
∈
N
≥1
]
determines
an
automorphism
of
the
monoid
M
,
hence
a
self-equivalence
of
C,
which
clearly
fails
to
preserve
“O
×
(−)”,
“O
(−)”
[i.e.,
the
subspaces
{0}
×
V
×
{0},
{0}
×
V
×
W
⊆
U
×
V
×
W
]
—
cf.
Theorem
3.4,
(iv).
Example
3.10.
Non-slim
Base
Categories.
Let
G
be
a
group,
whose
center
we
denote
by
Z(G);
D
a
one-object
category
whose
unique
object
has
endomorphism
monoid
G;
C
a
one-object
category
whose
unique
object
has
endomorphism
monoid
G
×
F.
Thus,
the
projection
G
×
F
→
G
determines
a
functor
C
→
D;
C
may
be
identified
with
the
elementary
Frobenioid
determined
by
the
[manifestly
non-
dilating]
monoid
on
D
that
assigns
to
the
unique
object
of
D
the
monoid
Z
≥0
and
to
every
morphism
of
D
the
identity
automorphism
of
Z
≥0
.
In
particular,
C
is
a
Frobenioid
of
Frobenius-normalized
and
isotropic
type,
which
is
not
of
group-like
type
[cf.
Proposition
1.5,
(i),
(ii)];
D
is
manifestly
of
FSM-,
hence
also
of
FSMFF-
type
[cf.
§0].
Thus,
C
is
of
standard
type.
On
the
other
hand,
if
α
:
F
→
Z(G)
is
any
nontrivial
homomorphism
of
monoids
that
factors
as
the
composite
of
the
natural
surjection
F
→
N
≥1
with
a
homomorphism
of
monoids
N
≥1
→
Z(G),
then
the
automorphism
of
monoids
∼
G
×
F
→
G
×
F
(g,
f
)
→
(g
·
α(f
),
f
)
∼
[where
g
∈
G,
f
∈
F]
determines
a
self-equivalence
C
→
C
which
clearly
fails
to
preserve
base-identity
endomorphisms
of
Frobenius
type
[cf.
Theorem
3.4,
(v)].
THE
GEOMETRY
OF
FROBENIOIDS
I
73
Finally,
before
proceeding,
we
consider
the
case
of
Frobenioids
of
group-like
type
in
a
bit
more
detail.
Proposition
3.11.
(Frobenioids
of
Isotropic,
Unit-trivial,
and
Group-
like
Type)
For
i
=
1,
2,
let
Φ
i
be
the
zero
monoid
[more
precisely:
any
functor
D
i
→
Mon
all
of
whose
values
are
monoids
of
cardinality
one]
on
a
connected,
totally
epimorphic
category
D
i
of
FSMFF-type;
C
i
→
F
Φ
i
a
Frobenioid
of
isotropic,
unit-trivial,
and
group-like
type;
∼
Ψ
:
C
1
→
C
2
an
equivalence
of
categories.
Then:
(i)
The
functor
C
i
→
F
Φ
i
is
an
equivalence
of
categories.
(ii)
Ψ
preserves
base-isomorphisms,
pull-back
morphisms,
linear
mor-
phisms,
and
morphisms
of
Frobenius
type.
(iii)
Suppose
that
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
base-identity
endomorphisms.
Then
there
exists
a
1-unique
functor
Ψ
Base
:
D
1
→
D
2
that
fits
into
a
1-commutative
diagram
C
1
⏐
⏐
D
1
Ψ
−→
Ψ
Base
−→
C
2
⏐
⏐
D
2
[where
the
vertical
arrows
are
the
natural
projection
functors;
the
horizontal
ar-
rows
are
equivalences
of
categories].
Finally,
if
D
1
,
D
2
are
slim,
then
each
of
the
composite
functors
of
this
diagram
is
rigid.
Proof.
First,
we
consider
assertion
(i).
By
Proposition
3.3,
(iii),
(iv),
the
functor
C
i
→
F
Φ
i
is
essentially
surjective
and
faithful.
Since
the
Frobenioid
C
i
is
of
group-
like
and
isotropic
type,
it
follows
that
every
pre-step
of
C
i
is
an
isomorphism
[cf.
Propositions
1.4,
(i);
1.8,
(iii)],
hence
that
the
Frobenioid
C
i
is
of
Aut-ample
and
base-trivial
[cf.
Definition
1.3,
(i),
(b)],
as
well
as
unit-trivial,
type.
Thus,
it
follows
from
Proposition
3.3,
(v),
that
the
functor
C
i
→
F
Φ
i
is
an
equivalence
of
categories.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
Observe
that
since
D
i
is
of
FSMFF-type,
it
follows
that
D
i
has
no
FSMI-endomorphisms
[cf.
§0],
hence
that
a
morphism
of
C
i
is
an
FSMI-endomorphism
if
and
only
if
it
is
a
prime-Frobenius
endomorphism
[cf.
Propositions
1.11,
(vi);
1.14,
(i);
the
evident
structure
of
F
Φ
i
].
Thus,
Ψ
preserves
the
prime-Frobenius
endomorphisms,
hence
also
prime-Frobenius
morphisms
[since
every
prime-Frobenius
morphism
is
abstractly
equivalent
to
a
prime-Frobenius
en-
domorphism].
But
this
implies
that
Ψ
preserves
the
morphisms
of
Frobenius
type
[cf.
Proposition
1.10,
(v)],
hence
also
the
linear
morphisms
[cf.
Proposition
1.7,
(iii)].
Since
the
[co-angular]
pre-steps
of
C
i
are
isomorphisms
[cf.
Proposition
1.8,
74
SHINICHI
MOCHIZUKI
(iii)],
it
thus
follows
that
Ψ
preserves
the
pull-back
morphisms
[cf.
Proposition
1.7,
(iii)],
as
well
as
the
base-isomorphisms
[cf.
Proposition
1.7,
(ii)].
This
completes
the
proof
of
assertion
(ii).
Finally,
we
consider
assertion
(iii).
Write
N
for
the
one-object
category
whose
unique
object
has
endomorphism
monoid
equal
to
N
≥1
.
Then
we
have
equivalences
of
categories
∼
∼
C
i
→
F
Φ
i
→
D
i
×
N
[cf.
assertion
(i)].
Moreover,
one
verifies
immediately
that
the
base-identity
endo-
∼
morphisms
of
C
i
are
precisely
the
endomorphisms
of
C
i
→
D
i
×
N
that
arise
from
elements
of
N
≥1
;
let
us
refer
to
such
endomorphisms
as
“N
≥1
-endomorphisms”.
Thus,
it
follows
from
our
assumption
concerning
the
preservation
of
base-identity
endomorphisms
that
the
N
≥1
-endomorphisms
are
preserved
by
Ψ.
Note,
more-
over,
that
D
i
may
be
reconstructed
from
C
i
by
considering
equivalence
classes
of
morphisms
of
C
i
,
where
two
morphisms
of
C
i
are
regarded
as
equivalent
if
they
admit
composites
with
an
N
≥1
-endomorphism
which
are
equal.
Thus,
we
obtain
a
1-commutative
diagram
as
in
the
statement
of
assertion
(ii).
Finally,
the
rigidity
assertion
in
the
statement
of
assertion
(ii)
follows
immediately
from
Proposition
1.13,
(i).
THE
GEOMETRY
OF
FROBENIOIDS
I
75
Section
4:
Category-theoreticity
of
the
Divisor
Monoid
In
the
present
§4,
we
show
that
the
monoid
on
the
base
category
that
appears
in
the
definition
of
a
Frobenioid
[cf.
Definition
1.3]
may,
under
suitable
conditions,
be
reconstructed
entirely
category-theoretically.
Together
with
the
results
of
§3,
this
allows
us
to
conclude,
under
suitable
conditions,
that
the
functor
to
an
elementary
Frobenioid
that
appears
in
the
definition
of
a
Frobenioid
[cf.
Definition
1.3]
may
be
recovered
entirely
from
the
structure
of
a
Frobenioid
as
an
abstract
category
[cf.
Corollary
4.11].
In
the
following
discussion,
we
maintain
the
notation
of
§1,
§2,
§3.
Also,
we
assume
that
we
have
been
given
a
divisorial
monoid
Φ
on
a
connected,
totally
epimorphic
category
D
and
a
Frobenioid
C
→
F
Φ
.
Proposition
4.1.
(Primary
Steps)
Suppose
further
that
C
is
of
perfect
and
isotropic
type,
and
that
Φ
is
perf-factorial.
Let
A
∈
Ob(C)
be
Div-Frobenius-
trivial;
φ
:
B
→
A,
ψ
:
A
→
C,
δ
:
D
→
E,
:
E
→
F,
ι
:
I
→
F
steps
of
C.
For
n
∈
N
≥1
,
let
α
n
∈
End
C
(A)
be
a
Div-identity
endomorphism
of
Frobenius
type
such
that
deg
Fr
(α
n
)
=
n.
Then:
(i)
φ
is
primary
if
and
only
if,
for
every
factorization
φ
=
φ
A
◦
φ
B
,
where
φ
B
:
B
→
B
,
φ
A
:
B
→
A
are
steps,
there
exists
a
commutative
diagram
B
φ
B
−→
φ
A
B
−→
⏐
⏐
β
B
ζ
−→
A
⏐
⏐
α
n
A
where
n
∈
N
≥1
;
β
is
a
morphism
of
Frobenius
type;
and
ζ
=
φ◦ζ
;
and
ζ
:
B
→
B
is
a
pre-step.
(ii)
Suppose
that
φ
is
primary.
Then
the
composite
ψ
◦
φ
:
B
→
C,
hence
also
ψ,
is
primary
if
and
only
if,
for
every
factorization
ψ
◦
φ
=
ψ
◦
φ
,
where
φ
:
B
→
A
,
ψ
:
A
→
C
are
steps,
there
exist
factorizations
φ
=
ζ
◦φ
,
φ
=
ζ
◦φ
,
where
φ
:
B
→
A
is
a
step,
and
ζ
:
A
→
A,
ζ
:
A
→
A
are
pre-steps.
(iii)
∗
(Div(
)),
ι
∗
(Div(ι))
∈
Φ(F
)
[where
we
write
∗
,
ι
∗
for
the
respective
bijections
induced
by
the
functor
Φ]
have
disjoint
supports
[cf.
Definition
2.4,
(i),
(d)]
if
and
only
if
every
pre-step
ζ
:
Z
→
F
such
that
there
exist
pre-steps
,
ι
satisfying
=
ζ
◦
,
ι
=
ζ
◦
ι
is,
in
fact,
an
isomorphism.
In
this
case,
we
shall
say
that
,
ι
are
co-primary.
If
,
ι
are
co-primary,
then
there
exists
a
cartesian
diagram
in
the
category
of
pre-steps
U
−→
⏐
⏐
ι
I
ι
−→
E
⏐
⏐
F
76
SHINICHI
MOCHIZUKI
such
that
∗
(
∗
(Div(
)))
=
ι
∗
(Div(ι)),
ι
∗
(ι
∗
(Div(ι
)))
=
mary,
then
so
are
,
ι
.
∗
(Div(
));
if
,
ι
are
pri-
(iv)
δ
is
primary
if
and
only
if
there
exists
a
p
∈
Prime(Φ(F
))
such
that
the
following
condition
holds:
For
every
primary
:
E
→
F
such
that
∗
(Div(
))
∈
p
[where
we
write
∗
for
the
bijection
induced
by
the
functor
Φ],
there
exists
a
factorization
=
◦ζ,
where
ζ
is
a
pre-step,
if
and
only
if
there
exists
a
factorization
◦
δ
=
◦
θ,
where
θ
is
a
pre-step.
(v)
is
primary
if
and
only
if
there
exists
a
p
∈
Prime(Φ(D))
such
that
the
following
condition
holds:
For
every
primary
δ
:
D
→
E
such
that
Div(δ
)
∈
p,
there
exists
a
factorization
δ
=
ζ
◦
δ
,
where
ζ
is
a
pre-step,
if
and
only
if
there
exists
a
factorization
◦
δ
=
θ
◦
δ
,
where
θ
is
a
pre-step.
Proof.
First,
we
consider
assertion
(i).
By
applying
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d),
to
the
various
pre-steps
over
A,
it
follows
def
that,
if
we
write
x
φ
=
φ
∗
(Div(φ))
∈
Φ(A)
[where
we
write
φ
∗
for
the
bijection
induced
by
the
functor
Φ],
then
the
condition
of
assertion
(i)
may
be
translated
into
the
language
of
monoids
as
follows:
For
every
equation
x
φ
=
x
A
+
x
B
in
Φ(A),
where
x
A
,
x
B
=
0,
we
have
x
φ
x
A
.
Now
the
equivalence
of
this
condition
with
the
condition
that
x
φ
is
primary
follows
immediately
from
the
definition
of
the
term
“primary”
[cf.
§0],
together
with
the
fact
that
Φ(A)
is
perfect
[cf.
Proposition
1.10,
(iii)].
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
Again,
we
apply
Definition
1.3,
(iii),
(d),
to
the
various
pre-steps
over
C,
to
obtain
the
following
translation
of
the
condition
of
as-
def
def
sertion
(ii)
into
the
language
of
monoids
[where
we
set
x
φ
=
ψ
∗
(φ
∗
(Div(φ))),
x
ψ
=
ψ
∗
(Div(ψ))
∈
Φ(C)]:
For
every
equation
x
φ
+
x
ψ
=
x
φ
+
x
ψ
in
Φ(A),
where
x
φ
,
x
ψ
=
0,
there
exists
a
0
=
x
φ
∈
Φ(A)
such
that
x
φ
≤
x
φ
,
x
φ
≤
x
φ
.
Now
the
necessity
of
this
condition
follows
immediately
from
the
structure
of
the
Φ(A)
p
,
where
p
∈
Prime(Φ(A))
[cf.
Definition
2.4,
(i),
(b)],
whereas
the
sufficiency
of
this
condition
follows
by
taking
x
φ
≤
x
ψ
[cf.
Definition
2.4,
(i),
(c),
(d);
the
fact
that
Φ(A)
is
perfect].
This
completes
the
proof
of
assertion
(ii).
Next,
we
consider
assertion
(iii).
By
applying
the
second
equivalence
of
cat-
egories
of
Definition
1.3,
(iii),
(d),
to
the
various
pre-steps
over
F
,
we
obtain
the
following
translation
of
the
condition
of
assertion
(iii)
into
the
language
of
monoids
def
def
[where
we
set
x
=
∗
(Div(
)),
x
ι
=
ι
∗
(Div(ι))
∈
Φ(F
)]:
Every
x
ζ
∈
Φ(F
)
such
that
x
ζ
≤
x
,
x
ζ
≤
x
ι
is,
in
fact,
equal
to
0.
THE
GEOMETRY
OF
FROBENIOIDS
I
77
The
necessity
and
sufficiency
of
this
condition
then
follow
immediately
by
consid-
ering
the
“primary
factorizations”
of
x
,
x
ι
[cf.
Definition
2.4,
(i),
(c),
(d);
the
fact
that
Φ(A)
is
perfect].
The
cartesian
diagram
[with
the
desired
properties]
then
follows
from
the
fact
that
“for
x
U
∈
Φ(F
),
x
+
x
ι
≤
x
U
if
and
only
if
x
≤
x
U
,
x
ι
≤
x
U
”
[cf.
Definition
2.4,
(i),
(c),
(d);
the
fact
that
Φ(A)
is
perfect].
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
This
time,
we
apply
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d),
to
the
various
pre-steps
over
F
,
to
obtain
the
following
translation
of
the
condition
of
assertion
(iv)
concerning
p
∈
Prime(Φ(F
))
def
into
the
language
of
monoids
[where
we
set
x
δ
=
Φ(F
)]:
∗
(δ
∗
((Div(δ))),
x
def
=
∗
(Div(
))
∈
/
p,
x
≤
x
if
and
only
if
x
≤
x
δ
+
x
.
For
every
primary
element
x
∈
The
necessity
and
sufficiency
of
this
condition
then
follow
immediately
by
com-
paring
the
“primary
factorizations”
of
x
,
x
δ
+
x
[cf.
Definition
2.4,
(i),
(c),
(d);
the
fact
that
Φ(A)
is
perfect].
Also,
we
observe
that
assertion
(v)
follows
by
an
entirely
similar
argument
obtained
by
“reversing
the
direction
of
the
arrows”.
This
completes
the
proof
of
assertions
(iv),
(v).
Theorem
4.2.
(Category-theoreticity
of
Primary
Steps)
For
i
=
1,
2,
let
Φ
i
be
a
perf-factorial
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D
i
;
C
i
→
F
Φ
i
a
Frobenioid
of
standard
and
isotropic
type,
which
is
not
of
group-like
type;
∼
Ψ
:
C
1
→
C
2
an
equivalence
of
categories.
Then:
(i)
Ψ
preserves
primary
steps,
Div-identity
endomorphisms,
Div-Frobenius-
trivial
objects,
and
universally
Div-Frobenius-trivial
objects.
(ii)
There
exists
a
unique
isomorphism
Ψ
Prime
between
the
functors
Ob(C
i
bs-iso
)
A
i
→
Prime(Φ
i
(A
i
))
[where
i
=
1,
2]
on
C
i
bs-iso
which
satisfies
the
following
property:
Suppose
that
A
2
=
Ψ(A
1
);
p
1
∈
Prime(Φ
1
(A
1
)),
p
2
∈
Prime(Φ
2
(A
2
))
correspond
under
Ψ
Prime
.
For
i
=
1,
2,
write
∼
{
A
i
(C
i
coa-pre
)}
p
i
→
Order(Φ
i
(A
i
)
p
i
);
∼
{(C
i
coa-pre
)
A
i
}
p
i
→
Order(Φ
i
(A
i
)
p
i
)
opp
for
the
respective
full
subcategories
and
restricted
equivalences
of
categories
deter-
mined
by
the
full
subcategory
Order(Φ
i
(A
i
)
p
i
)
⊆
Order(Φ
i
(A
i
))
78
SHINICHI
MOCHIZUKI
arising
from
the
submonoid
Φ
i
(A
i
)
p
i
⊆
Φ
i
(A
i
).
Then
the
map
induced
by
Ψ
on
pre-steps
[cf.
(i);
Theorem
3.4,
(ii)]
induces
equivalences
of
categories
∼
{
A
1
(C
1
coa-pre
)}
p
1
→
{
A
2
(C
2
coa-pre
)}
p
2
;
∼
{(C
1
coa-pre
)
A
1
}
p
1
→
{(C
2
coa-pre
)
A
2
}
p
2
hence
equivalences
of
categories
as
follows:
∼
Order(Φ
1
(A
1
)
p
1
)
→
Order(Φ
2
(A
2
)
p
2
);
∼
Order(Φ
1
(A
1
)
p
1
)
opp
→
Order(Φ
2
(A
2
)
p
2
)
opp
(iii)
If,
moreover,
in
the
situation
of
(ii),
the
A
i
are
Div-Frobenius-trivial,
then
the
last
two
equivalences
of
categories
of
(ii)
arise
from
isomorphisms
of
monoids
∼
∼
Φ
1
(A
1
)
p
1
→
Φ
2
(A
2
)
p
2
;
Φ
1
(A
1
)
p
1
→
Φ
2
(A
2
)
p
2
which
we
shall
refer
to,
respectively,
as
the
right-hand
and
left-hand
isomor-
phisms
induced
by
Ψ
[cf.
Example
4.3
below].
Proof.
First,
we
observe
that
by
Proposition
1.10,
(vi),
every
group-like
object
is
Frobenius-trivial,
hence,
in
particular,
Div-Frobenius-trivial;
moreover,
[by
the
definition
of
a
“group-like
object”]
every
endomorphism
of
a
group-like
object
is
a
Div-identity
endomorphism,
and
every
pre-step
to
or
from
a
group-like
object
is
an
isomorphism
[cf.
Propositions
1.4,
(i),
(iii);
1.8,
(iii)].
Thus,
since
Ψ
pre-
serves
non-group-like
objects
[cf.
Theorem
3.4,
(ii)]
and
pull-back
morphisms
[cf.
Theorem
3.4,
(iii)],
we
may
assume
for
the
remainder
of
the
proof
of
Theorem
4.2,
without
loss
of
generality,
that
the
objects
under
consideration
are
non-group-like.
Now
by
Proposition
1.14,
(v)
[cf.
also
Theorem
3.4,
(ii)],
it
follows
immediately
that
Ψ
preserves
non-group-like
Div-Frobenius-trivial
objects,
as
well
as
Div-identity
prime-Frobenius
endomorphisms
of
such
objects.
Since
Ψ
preserves
morphisms
of
Frobenius
type
and
Frobenius
degrees
[cf.
Theorem
3.4,
(iii)],
we
thus
conclude
that
to
complete
the
proof
of
assertion
(i),
it
suffices
to
prove
that
Ψ
preserves
primary
steps
and
Div-identity
endomorphisms.
Moreover,
to
prove
the
remainder
of
asser-
tion
(i)
[i.e.,
that
Ψ
preserves
primary
steps
and
Div-identity
endomorphisms]
and
assertions
(ii),
(iii),
clearly
it
suffices
to
do
so
after
passing
to
the
perfections
of
the
C
i
[cf.
Theorem
3.4,
(iii)];
thus,
for
the
remainder
of
the
proof
of
Theorem
4.2,
we
may
assume,
without
loss
of
generality,
that
the
C
i
are
of
perfect
type
[cf.
also
Proposition
5.5,
(iii),
below].
def
Now
let
A
1
∈
Ob(C
1
)
be
a
non-group-like
Div-Frobenius-trivial
object;
A
2
=
Ψ(A
1
).
Then
it
follows
formally
from
Proposition
4.1,
(i),
(ii)
[cf.
also
Theorem
3.4,
(ii),
(iii)]
that
Ψ
maps
primary
steps
to
or
from
A
1
to
primary
steps
to
or
from
A
2
in
such
a
way
that
primary
steps
B
1
→
A
1
,
A
1
→
C
1
with
primary
composite
B
1
→
C
1
are
mapped
to
primary
steps
B
2
→
A
2
,
A
2
→
C
2
with
primary
composite
B
2
→
C
2
.
Next,
let
A
1
→
F
1
be
a
primary
step.
Then
it
follows
immediately
from
Proposition
4.1,
(iii),
together
with
what
we
have
already
shown
concerning
primary
steps
to
or
from
A
1
,
that
Ψ
def
maps
primary
steps
to
or
from
F
1
to
primary
steps
to
or
from
F
2
=
Ψ(F
1
)
in
such
THE
GEOMETRY
OF
FROBENIOIDS
I
79
a
way
that
primary
steps
F
1
→
F
1
,
F
1
→
F
1
with
primary
composite
F
1
→
F
1
are
mapped
to
primary
steps
F
2
→
F
2
,
F
2
→
F
2
with
primary
composite
F
2
→
F
2
.
[Indeed,
to
see
this,
it
suffices
to
consider
the
following
two
situations
[depending
on
whether
the
primary
steps
A
1
→
F
1
,
F
1
→
F
1
are
co-primary
or
not]:
(a)
primary
steps
B
i
→
A
i
,
A
i
→
C
i
with
primary
composite
such
that
the
primary
steps
to
or
from
F
i
under
consideration
are
subordinate
to
the
primary
composite
B
i
→
C
i
;
(b)
commutative
diagrams
A
i
−→
F
i
⏐
⏐
⏐
⏐
A
i
⏐
⏐
−→
F
i
⏐
⏐
A
i
−→
F
i
[where
i
=
1,
2]
in
which
both
the
upper
and
lower
squares
are
cartesian
diagrams
as
in
Proposition
4.1,
(iii),
and
all
the
arrows
originating
from
A
i
,
as
well
as
the
vertical
composite
A
i
→
A
i
→
A
i
,
are
primary
steps.]
Next,
observe
that
for
a
suitable
choice
of
non-group-like
Div-Frobenius-trivial
A
1
[e.g.,
a
Frobenius-trivial
A
1
—
cf.
Definition
1.3,
(i),
(a),
(b)],
it
follows
that
for
any
object
C
1
∈
Ob(C
1
)
that
is
base-isomorphic
to
A
1
,
there
exist
pre-steps
B
1
→
C
1
,
B
1
→
A
1
.
Moreover,
observe
that
[by
applying
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d)]
any
primary
step
to
or
from
B
1
,
as
well
as
any
primary
composite
of
a
primary
step
to
B
1
with
a
primary
step
from
B
1
,
may
always
be
written
in
the
form
D
1
→
E
1
where
the
composite
D
1
→
E
1
→
F
1
of
the
above
arrow
D
1
→
E
1
with
some
arrow
E
1
→
F
1
factors
as
a
composite
D
1
→
B
1
→
A
1
→
F
1
in
which
D
1
→
B
1
is
a
pre-
step,
B
1
→
A
1
is
the
pre-step
introduced
above,
and
A
1
→
F
1
is
a
primary
step
[so
in
the
case
of
a
primary
step
from
B
1
,
D
1
=
B
1
;
in
the
case
of
a
primary
step
to
B
1
,
E
1
=
B
1
].
Thus,
by
applying
Proposition
4.1,
(iv)
[to
the
arrows
D
1
→
E
1
→
F
1
],
together
with
what
we
have
already
shown
concerning
primary
steps
to
or
from
F
1
,
we
conclude
that
Ψ
maps
primary
steps
to
or
from
B
1
to
primary
steps
to
or
def
from
B
2
=
Ψ(B
1
)
in
such
a
way
that
primary
steps
B
1
→
B
1
,
B
1
→
B
1
with
primary
composite
B
1
→
B
1
are
mapped
to
primary
steps
B
2
→
B
2
,
B
2
→
B
2
with
primary
composite
B
2
→
B
2
.
In
a
similar
vein,
we
observe
that
[by
applying
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d)]
a
primary
step
to
or
from
C
1
,
as
well
as
any
primary
composite
of
a
primary
step
to
C
1
with
a
primary
step
from
C
1
,
may
always
be
written
in
the
form
E
1
→
F
1
where
the
composite
D
1
→
E
1
→
F
1
of
some
arrow
D
1
→
E
1
with
the
above
arrow
E
1
→
F
1
factors
as
a
composite
D
1
→
B
1
→
C
1
→
F
1
in
which
D
1
→
B
1
is
a
primary
pre-step,
B
1
→
C
1
is
the
pre-step
introduced
above,
and
C
1
→
F
1
is
a
pre-step
[so
in
the
case
of
a
primary
step
from
C
1
,
E
1
=
C
1
;
in
the
case
of
80
SHINICHI
MOCHIZUKI
a
primary
step
to
C
1
,
F
1
=
C
1
].
Thus,
by
applying
Proposition
4.1,
(v)
[to
the
arrows
D
1
→
E
1
→
F
1
],
together
with
what
we
have
already
shown
concerning
primary
steps
to
or
from
D
1
[i.e.,
where
we
regard
“D
1
”
as
a
“sort
of
B
1
”,
which
is
possible
in
light
of
the
existence
of
the
composite
pre-step
D
1
→
B
1
→
A
1
],
we
conclude
that
Ψ
maps
primary
steps
to
or
from
C
1
to
primary
steps
to
or
from
def
C
2
=
Ψ(C
1
)
in
such
a
way
that
primary
steps
C
1
→
C
1
,
C
1
→
C
1
with
primary
composite
C
1
→
C
1
are
mapped
to
primary
steps
C
2
→
C
2
,
C
2
→
C
2
with
primary
composite
C
2
→
C
2
.
Since
C
1
was,
in
effect,
allowed
to
be
an
arbitrary
non-group-like
object
of
C
1
,
we
thus
conclude
that
Ψ
preserves
primary
steps.
Moreover,
by
thinking,
for
A
i
∈
Ob(C
i
)
[where
i
=
1,
2]
of
an
element
of
Prime(Φ
i
(A
i
))
as
an
equivalence
class
of
primary
steps
to
or
from
A
i
[where
the
correspondence
between
elements
of
Prime(Φ
i
(A
i
))
and
equivalence
classes
of
primary
steps
is
defined
by
“Div(−)”
—
cf.
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d)],
we
thus
obtain
that
Ψ
induces
a
bijection
∼
Ψ
Prime
(A
1
)
:
Prime(Φ
1
(A
1
))
→
Prime(Φ
2
(A
2
))
def
[where
A
2
=
Ψ(A
1
)]
as
well
as
corresponding
equivalences
of
categories
∼
{
A
1
(C
1
coa-pre
)}
p
1
→
{
A
2
(C
2
coa-pre
)}
p
2
;
∼
{(C
1
coa-pre
)
A
1
}
p
1
→
{(C
2
coa-pre
)
A
2
}
p
2
[where
p
1
∈
Prime(Φ
1
(A
1
)),
p
2
∈
Prime(Φ
2
(A
2
))
correspond
via
Ψ
Prime
(A
1
)].
To
check
the
functoriality
of
Ψ
Prime
(−)
with
respect
to
arbitrary
base-isomor-
phisms,
it
suffices
to
check
it
with
respect
to
morphisms
of
Frobenius
type
and
pre-
steps
[cf.
Proposition
1.7,
(ii)].
In
the
case
of
a
morphism
of
Frobenius
type
B
i
→
A
i
[where
i
=
1,
2],
the
desired
functoriality
follows
by
considering
commutative
diagrams
B
i
−→
B
i
⏐
⏐
⏐
⏐
A
i
−→
A
i
[cf.
Proposition
1.10,
(i)]
—
where
the
vertical
morphisms
are
morphisms
of
Frobe-
nius
type,
and
the
horizontal
morphisms
are
primary
steps.
In
the
case
of
a
pre-step
B
i
→
A
i
[where
i
=
1,
2],
the
desired
functoriality
follows
by
considering
a
commu-
tative
diagram
B
i
−→
B
i
⏐
⏐
⏐
⏐
C
i
⏐
⏐
−→
C
i
⏐
⏐
A
i
−→
A
i
—
where
all
of
the
morphisms
are
pre-steps;
all
of
the
horizontal
morphisms,
as
well
as
the
vertical
morphisms
and
composite
morphisms
of
the
upper
square,
are
either
THE
GEOMETRY
OF
FROBENIOIDS
I
81
isomorphisms
or
primary
steps;
either
the
vertical
morphisms
of
the
lower
square
are
isomorphisms,
or
the
lower
square
is
a
cartesian
diagram
as
in
Proposition
4.1,
(iii).
This
completes
the
proof
of
the
functoriality
of
Ψ
Prime
(−),
hence
of
assertion
(ii).
Next,
we
observe
that
Ψ
preserves
Div-identity
endomorphisms.
Indeed,
since
the
Φ
i
are
non-dilating,
it
follows
that
if
A
∈
Ob(C
i
)
[where
i
=
1,
2],
then
α
∈
End
C
i
(A)
is
a
Div-identity
endomorphism
if
and
only
if
α
admits
a
factorization
α
=
β
◦
γ,
where
β
:
B
→
A
is
a
pull-back
morphism,
and
γ
:
A
→
B
is
a
base-
isomorphism,
such
that
for
every
primary
step
A
→
A,
there
exists
a
commutative
diagram
A
−→
A
⏐
⏐
⏐
γ
⏐
γ
B
−→
⏐
⏐
B
⏐
⏐
β
A
A
β
−→
in
which
the
horizontal
morphisms
are
primary
steps;
the
upper
horizontal
mor-
phism
is
the
given
primary
step;
the
equivalence
classes
of
the
primary
steps
A
→
∼
A,
B
→
B
correspond
via
the
bijection
Prime(Φ
i
(γ))
:
Prime(Φ
i
(B))
→
Prime(Φ
i
(A))
[cf.
the
functoriality
of
Ψ
Prime
(−)];
β
is
a
pull-back
morphism
[cf.
Proposition
1.11,
(v)];
the
primary
steps
A
→
A,
A
→
A
determine
the
same
element
of
Prime(Φ
i
(A)).
This
completes
the
proof
of
assertion
(i).
Finally,
we
consider
assertion
(iii).
Thus,
we
assume
that
the
A
i
are
Div-
Frobenius-trivial.
By
considering
commutative
diagrams
of
the
form
B
i
−→
A
i
A
i
−→
C
i
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
B
i
−→
A
i
A
i
−→
C
i
—
where
the
vertical
morphisms
are
morphisms
of
Frobenius
type
[cf.
Proposition
1.10,
(i)],
the
morphisms
A
i
→
A
i
are
Div-identity
endomorphisms,
and
the
hori-
zontal
morphisms
are
primary
steps
—
it
follows
that
the
equivalences
of
categories
in
question
arise
from
bijections
of
sets
∼
Φ
1
(A
1
)
p
1
→
Φ
2
(A
2
)
p
2
;
∼
Φ
1
(A
1
)
p
1
→
Φ
2
(A
2
)
p
2
that
are
compatible
both
with
“≤”
and
with
multiplication
by
elements
of
N
≥1
.
In
light
of
the
well-known
structure
of
the
monoids
Q
≥0
,
R
≥0
[cf.
Definition
2.4,
(i),
(b)],
this
is
enough
to
conclude
that
these
bijections
of
sets
are,
in
fact,
isomor-
phisms
of
monoids,
as
desired.
This
completes
the
proof
of
assertion
(iii).
Example
4.3.
Independence
of
Right-hand
and
Left-hand
Isomor-
phisms.
As
the
following
example
shows,
the
right-hand
and
left-hand
isomor-
phisms
of
Theorem
4.2,
(iii),
do
not
necessarily
coincide
[cf.
Remark
4.9.1
below]:
82
SHINICHI
MOCHIZUKI
Let
D
be
a
one-morphism
category;
Φ
the
monoid
on
D
whose
value
on
the
unique
object
of
D
is
the
monoid
Q
≥0
.
Now
we
define
a
category
C
as
follows:
The
objects
of
C
are
the
elements
of
Q.
The
morphisms
a
→
b
of
C
from
an
object
a
∈
Q
to
an
object
b
∈
Q
are
the
elements
d
∈
N
≥1
such
that
d
·
a
≤
b;
composition
of
morphisms
is
defined
by
multiplication
of
elements
of
N
≥1
.
We
shall
refer
to
the
element
d
∈
N
≥1
determined
by
a
morphism
of
C
as
the
Frobenius
degree
of
the
morphism.
Thus,
we
obtain
a
natural
functor
C
→
F
Φ
by
assigning
to
a
morphism
φ
:
a
→
b
[where
a,
b
∈
Q]
the
zero
divisor
b
−
deg
Fr
(φ)
·
a
∈
Q
≥0
and
Frobenius
degree
deg
Fr
(φ)
∈
N
≥1
.
Since
C
is
clearly
connected
and
totally
epimorphic,
this
functor
determines
a
pre-Frobenioid
structure
on
C.
More-
over,
the
object
0
∈
Q
is
Frobenius-trivial;
φ
:
a
→
b
is
a
morphism
of
Frobenius
type
if
and
only
if
b
=
deg
Fr
(φ)
·
a;
φ
:
a
→
b
is
a
pre-step
if
and
only
if
deg
Fr
(φ)
=
1;
all
morphisms
of
C
are
base-isomorphisms;
all
pull-back
morphisms
of
C
are
iso-
morphisms;
all
“O
(−)”
of
C
are
trivial;
no
object
of
C
is
group-like.
Thus,
one
verifies
immediately
that
C
is
a
Frobenioid
of
isotropic
type.
Since
D
is
clearly
of
FSMFF-type,
and
Φ
is
non-dilating,
it
follows
that
C
is
also
of
standard
type,
over
a
slim
base
category
D.
Now
one
verifies
immediately
that
if
λ
∈
Q
>0
,
then
the
assignment,
for
a
∈
Q
≥0
,
a
→
a;
−a
→
−λ
·
a
determines
a
self-equivalence
of
categories
∼
Ψ
λ
:
C
→
C
that
preserves
Frobenius
degrees
[cf.
Theorem
3.4,
(iii)].
On
the
other
hand,
it
follows
immediately
from
the
construction
of
Ψ
λ
that
the
right-hand
isomorphism
of
Theorem
4.2,
(iii),
is
the
identity
on
Q
≥0
,
while
the
left-hand
isomorphism
of
Theorem
4.2,
(iii),
is
given
by
multiplication
by
λ
on
Q
≥0
.
In
order
to
proceed
further
toward
the
goal
of
“reconstructing
Φ
category-
theoretically
from
C”,
it
is
necessary
to
find
natural
conditions
on
the
Frobenioid
C
that
will
allow
us
to
rule
out
“pathologies”
of
the
sort
discussed
in
Example
4.3.
One
approach
to
doing
this
is
the
introduction
of
the
birationalization
of
a
Frobenioid,
as
follows.
Proposition
4.4.
(Birationalization
of
a
Frobenioid
I)
For
A,
B
∈
Ob(C),
write:
def
(A,
B)
=
lim
Hom
C
(A
,
B)
Hom
birat
C
−
→
coa-pre
(A
→A)∈C
A
where
the
inductive
limit
is
parametrized
by
[say,
some
small
skeletal
subcat-
coa-pre
egory
of
]
C
A
,
and
the
transition
morphism
induced
by
a
co-angular
pre-step
coa-pre
]
is
the
natural
morphism
Hom
C
(A
,
B)
→
A
→
A
[regarded
as
a
morphism
in
C
A
Hom
C
(A
,
B).
Then:
THE
GEOMETRY
OF
FROBENIOIDS
I
83
(i)
Composition
of
morphisms
in
C
determines
a
natural
composition
map
(A,
B)
×
Hom
birat
(B,
C)
→
Hom
birat
(A,
C)
Hom
birat
C
C
C
[where
A,
B,
C
∈
Ob(C)],
hence
a
category
C
birat
,
whose
objects
are
the
objects
of
”.
Moreover,
there
exists
a
natural
C
and
whose
morphisms
are
given
by
“Hom
birat
C
1-commutative
diagram
of
functors
C
⏐
⏐
−→
F
Φ
⏐
⏐
C
birat
−→
F
Φ
gp
−→
F
0
D
where
the
functors
between
elementary
Frobenioids
are
those
induced
by
the
natural
morphisms
of
monoids
Φ
→
Φ
gp
→
0
D
;
0
D
is
the
monoid
on
D
all
of
whose
values
on
objects
of
D
are
equal
to
the
monoid
with
one
element
[so
F
0
D
is
the
product
category
of
D
with
the
one-object
category
determined
by
the
monoid
N
≥1
].
(ii)
The
functor
C
birat
→
F
0
D
of
(i)
determines
a
structure
of
Frobenioid
of
group-like
type
on
C
birat
.
Moreover,
the
functor
C
→
C
birat
is
faithful.
In
particular,
for
every
A
∈
Ob(C)
with
image
A
birat
in
C
birat
,
the
functor
C
→
C
birat
determines
an
injection
of
groups
O
(A)
gp
→
O
×
(A
birat
).
We
shall
refer
to
the
functor
“O
×
(−)”
on
D
associated
to
the
Frobenioid
C
birat
[cf.
Proposition
2.2,
(ii),
(iii)]
as
the
rational
function
monoid
of
the
Frobenioid
C.
(iii)
There
exists
a
unique
subfunctor
of
groups
Φ
birat
⊆
Φ
gp
such
that
the
functor
C
birat
→
F
Φ
gp
of
(i)
factors
through
the
subcategory
F
Φ
birat
⊆
F
Φ
gp
determined
by
Φ
birat
,
and,
moreover,
the
resulting
functor
C
birat
→
F
Φ
birat
induces,
for
each
A
birat
∈
Ob(C
birat
),
a
surjection
O
×
(A
birat
)
Φ
birat
(A
birat
),
whose
kernel
is
the
image,
via
the
injection
O
(A)
gp
→
O
×
(A
birat
)
of
(ii),
of
O
×
(A)
⊆
O
(A)
gp
.
(iv)
A
morphism
of
C
maps
to
a(n)
co-angular
morphism
(respectively,
iso-
morphism;
morphism
of
Frobenius
type;
pull-back
morphism;
morphism
of
a
given
Frobenius
degree;
isometry;
pre-step;
base-isomorphism)
of
C
birat
if
and
only
if
it
is
a(n)
co-angular
morphism
(respectively,
co-angular
pre-
step;
co-angular
base-isomorphism;
co-angular
linear
morphism;
morphism
of
a
given
Frobenius
degree;
arbitrary
morphism;
pre-step;
base-isomorphism)
of
C.
A
morphism
of
C
birat
is
a
base-identity
endomorphism
if
and
only
if
arises
from
a
pair
(α
:
A
→
A;
φ
:
A
→
A),
where
α
is
a
co-angular
pre-step
in
the
in-
dexing
category
of
the
inductive
limit
defining
Hom
birat
(A,
A),
and
α
and
φ
are
C
base-equivalent.
An
object
of
C
maps
to
an
isotropic
object
of
C
birat
if
and
only
if
it
is
an
isotropic
object
of
C.
84
SHINICHI
MOCHIZUKI
Proof.
First,
we
consider
assertion
(i).
Given
morphisms
φ
:
A
→
B,
ψ
:
B
→
C
[in
C]
and
co-angular
pre-steps
α
:
A
→
A,
β
:
B
→
B
[in
C],
it
follows
from
Proposition
1.11,
(vii),
that
there
exists
a
commutative
diagram
φ
ψ
A
−→
B
−→
C
⏐
⏐
⏐
β
⏐
α
α
A
←−
A
φ
−→
B
where
α
[hence
also
α
◦
α
]
is
a
co-angular
pre-step.
Then
we
take
the
composite
of
(A,
B)
with
the
image
of
ψ
in
Hom
birat
(B,
C)
to
be
the
the
image
of
φ
in
Hom
birat
C
C
birat
image
of
ψ
◦
φ
in
Hom
C
(A,
C).
To
show
that
this
assignment
is
independent
of
the
choice
of
α
,
φ
,
it
suffices
to
consider
commutative
diagrams
α
∗
A
∗
−→
⏐
⏐
α
A
α
−→
φ
A
−→
B
⏐
⏐
⏐
⏐
β
α
A
φ
−→
B
[where
α
,
α
,
α
∗
are
co-angular
pre-steps]
and
to
observe
that
since
β
is
a
monomorphism
[cf.
Definition
1.3,
(v),
(a)],
the
fact
that
β
◦
φ
◦
α
∗
=
φ
◦
α
◦
α
∗
=
φ
◦
α
◦
α
=
β
◦
φ
◦
α
implies
that
φ
◦α
∗
=
φ
◦α
,
i.e.,
that
ψ
◦φ
,
ψ
◦φ
determine
the
same
element
(A,
C).
Also,
one
verifies
immediately
that
composite
of
morphisms
of
of
Hom
birat
C
Hom
birat
(−,
−)
is
associative.
This
completes
the
definition
of
the
category
C
birat
.
C
Then
by
assigning
to
the
pair
(α
:
A
→
A,
φ
:
A
→
B)
the
element
Φ(α)
−1
{Div(φ
)
−
deg
Fr
(φ
)
·
Div(α)}
∈
Φ(A)
gp
[cf.
Remark
1.1.1]
one
verifies
immediately
that
the
functor
C
→
F
Φ
induces
a
functor
C
birat
→
F
Φ
gp
,
as
well
as
a
1-commutative
diagram
as
in
the
statement
of
assertion
(i).
This
completes
the
proof
of
assertion
(i).
Next,
we
observe
that
it
follows
formally
from
the
definition
of
C
birat
that
C
birat
is
connected;
moreover,
[cf.
the
discussion
of
the
composition
of
arrows
of
C
birat
in
the
proof
of
assertion
(i)]
the
total
epimorphicity
of
C
birat
follows
immediately
from
that
of
C.
Thus,
the
functor
C
birat
→
F
0
D
determines
a
structure
of
pre-Frobenioid
on
C
birat
.
Now
the
portion
of
assertion
(iv)
concerning
morphisms
of
a
given
Frobe-
nius
degree,
isometries
[cf.
the
monoid
structure
of
the
monoid
0
D
!],
pre-steps,
base-isomorphisms,
and
base-identity
endomorphisms
of
C
birat
follows
immediately
from
the
definitions.
The
portion
of
assertion
(iv)
concerning
co-angular
pre-steps
(−,
−)”;
Proposition
1.7,
of
C
follows
immediately
from
the
definition
of
“Hom
birat
C
(v)
[for
co-angular
pre-steps].
To
verify
the
portion
of
assertion
(iv)
concerning
co-angular
morphisms,
we
reason
as
follows:
Given
a
morphism
A
birat
→
B
birat
in
C
birat
,
any
factorization
THE
GEOMETRY
OF
FROBENIOIDS
I
85
A
birat
→
C
birat
→
D
birat
→
B
birat
in
C
birat
,
where
either
A
birat
→
C
birat
or
D
birat
→
B
birat
is
a
base-isomorphism,
C
birat
→
D
birat
is
a(n)
[isometric]
pre-
step,
and
D
birat
→
B
birat
is
linear,
arises
[cf.
the
proof
of
assertion
(i)]
from
a
factorization
A
→
C
→
D
→
B
in
C,
where
either
A
→
C
or
D
→
B
is
a
base-isomorphism,
C
→
D
is
a
pre-step,
and
D
→
B
is
linear.
Thus,
if
A
→
B
is
co-angular,
then
[by
applying
the
factorization
of
Definition
1.3,
(v),
(b),
to
C
→
D
,
we
conclude
that]
C
→
D
is
a
co-angular
pre-step,
so
C
birat
→
D
birat
is
an
isomorphism;
in
particular,
it
follows
that
A
birat
→
B
birat
is
co-angular.
On
the
other
hand,
if
A
birat
→
B
birat
is
co-angular,
then
C
birat
→
D
birat
is
an
isomorphism,
which
[by
the
portion
of
assertion
(iv)
concerning
isomorphisms
of
C
birat
]
implies
that
C
→
D
is
a
co-angular
pre-step,
hence
an
isomorphism
whenever
it
is
an
isometry
[cf.
Proposition
1.4,
(iii)];
thus,
A
→
B,
hence
also
any
morphism
A
→
B
appearing
in
a
factorization
A
→
A
→
B
[where
A
→
A
is
a
co-angular
pre-step],
is
co-angular.
The
portion
of
assertion
(iv)
concerning
morphisms
of
Frobenius
type
now
follows
formally
from
the
portion
of
assertion
(iv)
concerning
co-angular
morphisms,
isometries,
and
base-isomorphisms.
Next,
let
us
observe
that
it
is
immediate
from
the
definition
of
a
pull-back
morphism
[cf.
Definition
1.2,
(ii)]
that
any
pull-back
morphism
of
C
maps
to
a
pull-back
morphism
of
C
birat
.
Since,
moreover,
a
morphism
of
C
is
a
co-angular
linear
morphism
if
and
only
if
it
is
a
composite
of
a
co-angular
pre-step
and
a
pull-back
morphism
[cf.
Propositions
1.4,
(iv);
1.7,
(iii)],
it
thus
follows
[cf.
the
portion
of
assertion
(iv)
concerning
co-angular
pre-steps
of
C]
that
every
co-angular
linear
morphism
of
C
maps
to
a
pull-back
morphism
of
C
birat
.
On
the
other
hand,
if
φ
:
A
→
B
is
a
morphism
of
C
that
maps
to
a
pull-back
morphism
φ
birat
:
A
birat
→
B
birat
of
C
birat
,
then
it
follows
that
φ
is
linear,
hence
that
it
factors
as
a
composite
γ
◦
α,
where
α
:
A
→
C
is
a
pre-step,
and
γ
:
C
→
B
is
a
pull-back
morphism
[cf.
Proposition
1.7,
(iii)].
Thus,
we
obtain
an
equation
φ
birat
=
γ
birat
◦
α
birat
in
C
birat
,
where
φ
birat
,
γ
birat
are
pull-back
morphisms,
and
α
birat
is
a
base-isomorphism;
but
[by
the
isomorphism
of
functors
appearing
in
the
definition
of
a
“pull-back
morphism”
in
Definition
1.2,
(ii)]
this
implies
formally
that
α
birat
is
an
isomorphism,
hence
[by
the
portion
of
assertion
(iv)
concerning
co-angular
pre-steps
of
C]
that
α
is
a
co-angular
pre-step,
as
desired.
Finally,
the
portion
of
assertion
(iv)
concerning
isotropic
objects
follows
immediately
from
the
portion
of
assertion
(iv)
concerning
pre-steps
and
co-angular
pre-steps;
Proposition
1.4,
(i),
(iii);
Proposition
1.9,
(iv).
This
completes
the
proof
of
assertion
(iv).
In
light
of
the
“dictionary”
provided
by
assertion
(iv)
[cf.
also
Proposition
1.4,
(iv);
the
equivalence
of
categories
of
Proposition
1.9,
(ii)],
it
is
now
a
routine
exercise
to
check
that
C
birat
is,
in
fact,
a
Frobenioid
of
group-like
type.
Moreover,
it
is
immediate
from
the
definitions
[and
the
total
epimorphicity
of
C]
that
the
functor
C
→
C
birat
is
faithful
and
determines
an
injection
O
(A)
gp
→
O
×
(A
birat
),
for
A
∈
Ob(C).
This
completes
the
proof
of
assertion
(ii).
Now
assertion
(iii)
follows
immediately
from
the
existence
of
the
functor
C
birat
→
F
Φ
gp
of
assertion
(i)
[cf.
also
Proposition
1.5,
(ii)];
here,
we
note
that
the
computation
of
the
kernel
of
the
surjection
of
assertion
(iii)
follows
from
Definition
1.3,
(vi).
86
SHINICHI
MOCHIZUKI
Definition
4.5.
(i)
We
shall
say
that
an
object
of
C
is
birationally
Frobenius-normalized
if
its
image
in
C
birat
is
Frobenius-normalized.
[Thus,
any
birationally
Frobenius-
normalized
object
of
C
is
Frobenius-normalized
—
cf.
Proposition
4.4,
(ii),
(iv).]
If
every
object
of
C
is
birationally
Frobenius-normalized,
then
we
shall
say
that
C
is
of
birationally
Frobenius-normalized
type.
If
C
is
of
pre-model
and
birationally
Frobenius-normalized
type,
then
we
shall
say
that
C
is
of
model
type.
(ii)
Suppose
that
Φ
is
perf-factorial;
A
∈
Ob(C).
Then
we
shall
say
that
A
is
strictly
rational
if,
for
every
prime
p
∈
Prime(Φ(A)),
there
exists
an
element
/
Supp(b)
[cf.
a
−
b
∈
Φ
birat
(A),
where
a,
b
∈
Φ(A)
such
that
p
∈
Supp(a),
p
∈
Definition
2.4,
(i),
(d)].
We
shall
say
that
A
is
rational
if
there
exists
a
pull-back
morphism
B
→
A
in
C,
where
B
is
strictly
rational.
If
[Φ
is
perf-factorial,
and]
every
object
of
C
is
rational
(respectively,
strictly
rational),
then
we
shall
say
that
C
is
of
rational
(respectively,
strictly
rational)
type.
(iii)
We
shall
say
that
C
is
of
rationally
standard
type
if
the
following
conditions
are
satisfied:
(a)
C
is
of
birationally
Frobenius-normalized,
rational,
and
standard
type;
(b)
(C
un-tr
)
birat
admits
a
Frobenius-compact
object.
(iv)
We
shall
say
that
D
is
Div-slim
[relative
to
Φ]
if,
for
every
A
∈
Ob(D),
the
homomorphism
Aut(D
A
→
D)
→
Aut(D
A
→
Mon)
[induced
by
composition
with
the
functor
Φ
:
D
→
Mon]
is
injective.
[Thus,
if
D
is
slim,
then
it
is
Div-slim.]
Remark
4.5.1.
We
observe
in
passing
that
it
is
immediate
from
the
definitions
that
if
C
is
of
rationally
standard
type
(respectively,
of
standard
type),
then
so
is
C
istr
.
Example
4.6.
Frobenius-normalized
vs.
Birationally
Frobenius-normal-
ized.
As
the
following
example
shows,
it
is
not
necessarily
the
case
that
a
Frobe-
nioid
of
Frobenius-normalized
type
is
of
birationally
Frobenius-normalized
type:
Let
G
be
an
abelian
group,
written
additively.
For
each
p
∈
Primes,
let
ξ
p
∈
G.
def
Then
if
we
write
M
=
G
×
Z
×
Z,
then
the
assignment
M
(g,
a,
b)
→
(p
·
g
+
a
·
ξ
p
,
p
·
a,
p
·
b)
∈
M
determines
an
endomorphism
α
p
∈
End(M
)
of
the
module
M
such
that
α
p
com-
mutes
with
all
α
p
,
for
p
∈
Primes.
Thus,
we
obtain
a
homomorphism
N
≥1
→
End(M
),
i.e.,
an
action
of
N
≥1
on
M
;
write
α
n
for
the
image
in
End(M
)
of
n
∈
N
≥1
.
Write
N
for
the
monoid
whose
underlying
set
is
equal
to
the
direct
product
M
×
N
≥1
THE
GEOMETRY
OF
FROBENIOIDS
I
87
and
whose
monoid
structure
is
given
as
follows:
If
λ,
μ
∈
M
;
l,
m
∈
N
≥1
,
then
(λ,
l)
·
(μ,
m)
=
(λ
+
α
l
(μ),
l
·
m).
Now
let
D
be
a
one-morphism
category;
Φ
the
monoid
on
D
whose
unique
value
is
given
by
Z
≥0
×
Z
≥0
.
Let
C
be
the
category
whose
objects
A
n
are
indexed
by
elements
n
∈
Z,
and
whose
morphisms
A
n
1
→
A
n
2
[where
n
1
,
n
2
∈
Z]
consist
of
elements
(g,
a,
b,
d)
∈
N
such
that
a
≥
0,
b
≥
0,
n
2
−d·n
1
=
a+b;
composition
of
morphisms
is
determined
by
the
product
structure
of
N
.
The
assignment
(g,
a,
b,
d)
→
(a,
b,
d)
then
determines
a
functor
C
→
F
Φ
[which
lies
over
D].
Moreover,
one
checks
immediately
that,
relative
to
this
last
functor,
C
is
a
Frobenioid
of
isotropic
and
standard
type
which
is
not
of
group-like
type.
Also,
we
observe
that
the
object
A
0
∈
Ob(C)
is
Frobenius-trivial,
and
that
for
every
A
∈
Ob(C),
O
×
(A)
=
O
(A)
=
G.
On
the
other
hand,
one
computes
easily
that
for
A
birat
∈
Ob(C
birat
),
O
×
(A
birat
)
=
M
0
,
where
we
write
M
0
⊆
M
for
the
subgroup
of
(g,
a,
b)
∈
M
such
that
a
+
b
=
0.
Moreover,
the
morphisms
)
(0,
0,
0,
d)
∈
N
determine
a
homomorphism
N
≥1
→
End
C
(A
0
)
→
End
C
birat
(A
birat
0
[where
we
write
use
the
superscript
“birat”
to
denote
the
image
of
objects
of
C
in
C
birat
],
hence
an
action
of
N
≥1
on
O
×
(A
birat
)
=
M
0
,
which
is
easily
verified
to
coincide
with
the
restriction
to
M
0
of
the
original
action
of
N
≥1
on
M
.
Now
observe
that
C
is
of
[strictly]
rational
type
[cf.
Definition
4.5,
(ii)],
and,
moreover,
every
object
of
(C
un-tr
)
birat
is
Frobenius-compact.
On
the
other
hand,
if
the
ξ
p
=
0
[so
α
p
does
not
act
on
M
0
by
multiplication
by
p],
then
C
fails
to
be
of
birationally
Frobenius-normalized
type.
[In
a
similar
vein,
we
note
that
although
C
birat
is
“very
similar”
to
an
elementary
Frobenioid,
the
presence
of
the
“ξ
p
’s”
means
that
it
is
not,
in
general,
an
elementary
Frobenioid.]
Example
4.7.
Frobenius-slim
vs.
Div-slim.
(i)
Suppose
that
the
functor
Φ
:
D
→
Mon
maps
every
automorphism
of
D
to
an
identity
automorphism
of
Mon.
Then
it
follows
formally
that
D
is
Div-slim
if
and
only
if
D
is
slim.
In
particular,
if,
for
instance,
D
is
a
one-object
category,
A
∈
Ob(D),
and
End
D
(A)
is
a
nontrivial
residually
finite
group
G,
then
Aut(D
A
→
D)
=
Ker(Aut(D
A
→
D)
→
Aut(D
A
→
Mon))
=
G
—
so
[cf.
Remark
3.1.2]
D
is
Frobenius-slim,
but
not
Div-slim.
def
def
def
(ii)
Let
V
=
Q;
N
=
(N
≥1
)
gp
;
G
=
V
N
,
where
N
(⊆
Q)
acts
on
V
multi-
plicatively.
Let
D
be
a
one-object
category,
A
∈
Ob(D);
suppose
that
End
D
(A)
=
G
[so
Aut(D
A
→
D)
=
G].
Then
clearly
there
exists
an
injection
F
→
G,
so
D
fails
to
be
Frobenius-slim.
On
the
other
hand,
if
Φ
:
D
→
Mon
is
the
functor
determined
by
the
monoid
Z
≥0
g∈G
[i.e.,
the
copies
of
Z
≥0
are
indexed
by
the
elements
of
G]
equipped
with
the
G-action
obtained
by
letting
G
act
by
left
multiplication
on
the
indices
of
the
copies
of
Z
≥0
,
then
the
natural
map
Aut(D
A
→
D)
=
G
→
Aut(D
A
→
Mon)
88
SHINICHI
MOCHIZUKI
is
clearly
injective,
so
D
is
Div-slim
[relative
to
Φ].
Proposition
4.8.
(Birationalization
of
a
Frobenioid
II)
(i)
If
C
is
of
isotropic
type,
then
so
is
C
birat
.
(ii)
If
C
is
of
perfect
and
isotropic
type,
then
so
is
C
birat
.
(iii)
If
C
is
of
rationally
standard
type,
then
(C
istr
)
birat
is
of
standard
type.
(iv)
If
C
is
of
isotropic
and
pre-model
type,
then
so
is
C
birat
.
Proof.
Assertion
(i)
follows
formally
from
Proposition
4.4,
(iv).
To
prove
assertion
(ii),
observe
that
the
naive
Frobenius
functor
[cf.
Proposition
2.1]
determines
a
natural
1-commutative
diagram
[cf.
Proposition
4.4,
(ii),
(iv)]
Ψ
C
⏐
⏐
−→
C
birat
Ψ
birat
−→
C
⏐
⏐
C
birat
in
which
the
vertical
arrows
are
the
natural
functor
C
→
C
birat
of
Proposition
4.4,
(i);
the
horizontal
arrows
are
the
“naive
Frobenius
functor”
of
Proposition
2.1;
the
upper
horizontal
arrow
is
an
equivalence
of
categories
[by
our
assumption
that
C
is
of
perfect
type;
Proposition
2.1,
(iii)].
Since,
moreover,
Ψ
and
any
quasi-inverse
to
Ψ
preserve
[necessarily
co-angular,
since
C
is
of
isotropic
type]
pre-steps,
it
thus
follows
immediately
[cf.
the
definition
of
“C
birat
”]
that
Ψ
birat
is
also
an
equivalence
of
categories.
But
this
implies
[cf.
Proposition
2.1,
(iii)]
that
C
birat
is
of
perfect
type,
as
desired.
In
light
of
assertion
(i),
this
completes
the
proof
of
assertion
(ii).
Finally,
assertion
(iii)
follows
formally
from
the
definitions
[cf.
also
assertion
(i)];
assertion
(iv)
follows
formally
Proposition
4.4,
(iv)
[cf.
also
assertion
(i)].
We
are
now
ready
to
“reconstruct
Φ
category-theoretically
from
C”.
Theorem
4.9.
(Category-theoreticity
of
Divisor
Monoids)
For
i
=
1,
2,
let
Φ
i
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D
i
;
C
i
→
F
Φ
i
a
Frobenioid
of
rationally
standard
type;
∼
Ψ
:
C
1
→
C
2
an
equivalence
of
categories.
Then
there
exists
an
isomorphism
of
functors
∼
Ψ
Φ
:
Φ
1
→
Φ
2
[where
we
regard,
for
i
=
1,
2,
the
functor
Φ
i
:
D
i
→
Mon
as
a
functor
on
C
i
,
by
restriction
via
the
natural
projection
functor
C
i
→
D
i
]
lying
over
Ψ,
which
is
THE
GEOMETRY
OF
FROBENIOIDS
I
89
compatible
[when
the
C
i
are
of
isotropic,
but
not
of
group-like
type]
with
the
isomorphism
Ψ
Prime
of
Theorem
4.2,
(ii).
Proof.
First,
we
observe
[cf.
Theorem
3.4,
(i),
(ii)]
that
we
may
assume
without
loss
of
generality
that
C
1
,
C
2
are
of
isotropic
type
[cf.
Remark
4.5.1],
but
not
of
group-like
type
[since
Theorem
4.9
is
vacuous
if
C
1
,
C
2
are
of
group-like
type].
Next,
I
claim
that
to
complete
the
proof
of
Theorem
4.9,
it
suffices
to
show
that
the
right-hand
and
left-hand
isomorphisms
of
Theorem
4.2,
(iii),
coincide
[cf.
Re-
mark
4.9.1
below],
for
all
universally
Div-Frobenius-trivial
objects
[e.g.,
Frobenius-
trivial
objects
—
cf.
Remark
1.11.1].
Indeed,
if
the
right-hand
and
left-hand
iso-
morphisms
of
Theorem
4.2,
(iii),
coincide
for
all
universally
Div-Frobenius-trivial
objects,
then
it
follows
immediately
from
the
construction
of
the
isomorphism
of
functors
Ψ
Prime
in
the
proof
of
Theorem
4.2,
(ii),
that
Ψ
Prime
extends,
for
A
i
∈
Ob(C
i
bs-iso
),
p
i
∈
Prime(Φ
i
(A
i
))
[where
i
=
1,
2]
such
that
A
2
=
Ψ(A
1
),
p
2
=
Ψ
Prime
(p
1
),
to
an
isomorphism
of
monoids
∼
pf
Φ
1
(A
1
)
pf
p
1
→
Φ
2
(A
2
)
p
2
which
is
functorial
in
A
1
[regarded
as
an
object
of
C
1
bs-iso
].
Thus,
by
allowing
the
p
i
to
vary,
we
obtain,
for
A
i
∈
Ob(C
i
bs-iso
)
[where
i
=
1,
2]
such
that
A
2
=
Ψ(A
1
),
an
isomorphism
of
monoids
∼
pf
Φ
1
(A
1
)
pf
factor
→
Φ
2
(A
2
)
factor
[cf.
Definition
2.4,
(i),
(c)]
which
is
functorial
in
A
1
[regarded
as
an
object
of
C
1
bs-iso
].
Moreover,
by
applying,
say,
the
first
equivalence
of
categories
of
Definition
1.3,
(iii),
(d),
to
obtain
pre-steps
φ
:
A
→
B
of
C
i
with
arbitrary
prescribed
zero
divisor
and
considering
primary
steps
ψ
:
A
→
C
such
that
φ
=
ζ
◦
ψ
for
some
pre-step
ζ,
one
concludes
immediately
that
this
subset
maps
the
subset
Φ
1
(A
1
)
⊆
Φ
1
(A
1
)
pf
factor
[cf.
pf
Definition
2.4,
(i),
(c)]
onto
the
subset
Φ
2
(A
2
)
⊆
Φ
2
(A
2
)
factor
,
hence
determines
an
isomorphism
of
monoids
∼
Φ
1
(A
1
)
→
Φ
2
(A
2
)
which
is
functorial
in
A
1
[regarded
as
an
object
of
C
1
bs-iso
].
Finally,
the
functoriality
of
this
isomorphism
of
monoids
with
respect
to
pull-back
morphisms
follows
imme-
diately
by
“pulling
back
pre-steps”,
as
in
Proposition
1.11,
(v).
This
completes
the
proof
of
the
claim.
To
prove
that
the
right-hand
and
left-hand
isomorphisms
of
Theorem
4.2,
(iii),
coincide
for
all
universally
Div-Frobenius-trivial
objects,
we
reason
as
follows.
First
of
all,
by
passing
to
perfections
[cf.
Theorem
3.4,
(iii)],
we
may
assume
without
loss
of
generality
that
C
1
,
C
2
are
of
perfect
type
[cf.
also
Proposition
5.5,
(iii),
below].
Let
A
be
a
universally
Div-Frobenius-trivial
object
of
C
i
[where
i
=
1,
2].
Since
the
right-hand
and
left-hand
isomorphisms
of
Theorem
4.2,
(iii),
are
clearly
compatible
with
pull-back
morphisms
[cf.
Proposition
1.11,
(v);
the
proof
of
Theorem
4.2,
(iii)],
and
Ψ
preserves
pull-back
morphisms
[cf.
Theorem
3.4,
(iii)],
it
follows
that
we
may
assume
without
loss
of
generality
that
A
is
strictly
rational.
Let
us
refer
to
pairs
90
SHINICHI
MOCHIZUKI
of
primary
steps
β
:
A
→
B,
γ
:
C
→
A
such
that
Div(β)
=
(Φ
i
(γ))
−1
(Div(γ))
as
twin-primary
steps.
Then,
it
suffices
to
show,
for
each
p
∈
Prime(Φ
1
(A)),
the
existence
of
twin-primary
steps
with
zero
divisor
in
p
that
are
mapped
by
Ψ
to
twin-primary
steps
of
C
2
.
On
the
other
hand,
since
A
is
strictly
rational,
it
follows
[cf.
Definition
4.5,
(ii)]
that
there
exist,
for
each
p
∈
Prime(Φ
i
(A)),
cartesian
commutative
diagrams
of
pre-steps
as
in
Proposition
4.1,
(iii),
γ
γ
C
−→
D
⏐
⏐
⏐
⏐
γ
C
−→
D
⏐
⏐
⏐
⏐
γ
δ
B
β
−→
A
A
δ
α
−→
F
def
in
which
α,
β
are
twin-primary
with
zero
divisor
in
p;
the
pre-steps
ζ
=
β
◦
γ
:
C
→
A,
γ
:
C
→
A
are
Div-equivalent
[e.g.,
base-equivalent].
[Indeed,
Definition
4.5,
(ii)
[cf.
also
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d)],
guarantees
the
existence
of
base-equivalent
pre-steps
ζ,
γ
—
which
may,
moreover,
be
taken
to
be
co-primary
[cf.
Proposition
4.1,
(iii);
Definition
2.4,
(i),
(c),
(d)],
by
our
assumption
that
C
i
is
of
perfect
type
—
such
that
ζ
admits
a
factorization
β
◦
γ,
where
β
is
primary
with
zero
divisor
that
maps
via
Φ(β)
−1
to
an
element
of
p,
and
[again
by
our
assumption
that
C
i
is
of
perfect
type]
p
is
not
contained
in
the
support
of
(Φ(ζ))
−1
(Div(γ)).]
Conversely,
given
any
pair
of
cartesian
diagrams
of
pre-steps
as
in
Proposition
4.1,
(iii),
γ
γ
C
−→
D
⏐
⏐
⏐
γ
⏐
C
−→
D
⏐
⏐
⏐
⏐
γ
δ
B
β
−→
A
A
δ
α
−→
F
def
in
which
α,
β
are
primary
with
zero
divisor
in
p;
the
pre-steps
ζ
=
β
◦
γ
:
C
→
A,
γ
:
C
→
A
are
Div-equivalent
[e.g.,
base-equivalent],
it
follows
immediately
that
α,
β
are
twin-primary.
On
the
other
hand,
since
Ψ
preserves
pre-steps
[cf.
Theorem
3.4,
(ii)],
primary
steps
[cf.
Theorem
4.2,
(i)],
Div-equivalent
pairs
of
base-
isomorphisms
[cf.
Theorem
4.2,
(ii);
the
fact
that
Φ
i
is
non-dilating],
and
cartesian
diagrams
as
in
Proposition
4.1,
(iii)
[cf.
Proposition
4.1,
(iii),
or,
alternatively,
Theorem
4.2,
(ii)],
we
thus
conclude
that
for
each
p
∈
Prime(Φ
1
(A)),
there
exist
twin-primary
steps
with
zero
divisor
in
p
that
are
mapped
by
Ψ
to
twin-primary
steps
of
C
2
.
This
completes
the
proof
of
Theorem
4.9.
Remark
4.9.1.
One
verifies
immediately
that
the
Frobenioid
of
Example
4.3
is
not
of
rational
type.
Corollary
4.10.
(Category-theoreticity
of
the
Birationalization)
For
i
=
1,
2,
let
Φ
i
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D
i
of
FSMFF-type;
C
i
→
F
Φ
i
a
Frobenioid
of
quasi-isotropic
type;
∼
Ψ
:
C
1
→
C
2
THE
GEOMETRY
OF
FROBENIOIDS
I
91
an
equivalence
of
categories.
Then
there
exists
a
1-unique
functor
Ψ
birat
:
C
1
birat
→
C
2
birat
that
fits
into
a
1-commutative
diagram
Ψ
C
1
⏐
⏐
−→
C
1
birat
Ψ
birat
−→
C
2
⏐
⏐
C
2
birat
[where
the
vertical
arrows
are
the
natural
functors
of
Proposition
4.4,
(i);
the
hor-
izontal
arrows
are
equivalences
of
categories].
Finally,
if
D
1
,
D
2
are
slim,
and
C
1
,
C
2
are
of
birationally
Frobenius-normalized
type,
then
each
of
the
composite
functors
of
this
diagram
is
rigid.
Proof.
The
existence
and
1-uniqueness
of
a
1-commutative
diagram
as
in
the
statement
of
Corollary
4.10
follows
immediately
from
the
definition
of
“C
i
birat
”
[cf.
Proposition
4.4,
(i)],
and
the
fact
that
Ψ
preserves
co-angular
pre-steps
[cf.
Theorem
3.4,
(ii)].
The
rigidity
assertion
then
follows
immediately
from
Proposition
1.13,
(i),
by
considering
base-identity
endomorphisms
of
Frobenius
type
of
Frobenius-trivial
objects
of
C
i
,
under
the
hypothesis
that
the
C
i
are
birationally
Frobenius-normalized
[cf.,
e.g.,
the
proof
of
the
rigidity
assertion
of
Theorem
3.4,
(i)].
Corollary
4.11.
(Category-theoreticity
of
the
Functor
to
an
Elementary
Frobenioid
I)
For
i
=
1,
2,
let
Φ
i
be
a
perf-factorial
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D
i
which
is
Div-slim
[with
respect
to
Φ
i
];
C
i
→
F
Φ
i
a
Frobenioid
of
standard
type;
∼
Ψ
:
C
1
→
C
2
an
equivalence
of
categories.
If
C
1
,
C
2
are
of
group-like
type,
then
we
also
assume
that
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
base-isomorphisms.
Then:
(i)
There
exists
a
1-unique
functor
Ψ
un-tr
:
C
1
un-tr
→
C
2
un-tr
that
fits
into
a
1-commutative
diagram
Ψ
istr
−→
C
2
istr
C
1
istr
⏐
⏐
⏐
⏐
C
1
un-tr
Ψ
un-tr
−→
C
2
un-tr
[where
the
vertical
arrows
are
the
natural
projection
functors;
the
horizontal
arrows
are
equivalences
of
categories;
Ψ
istr
is
the
restriction
of
Ψ
to
C
1
istr
—
cf.
Theorem
3.4,
(i)].
Moreover,
each
of
the
composite
functors
of
this
diagram
is
rigid.
(ii)
There
exists
a
1-unique
functor
Ψ
Base
:
D
1
→
D
2
that
fits
into
a
1-
commutative
diagram
Ψ
C
1
−→
C
2
⏐
⏐
⏐
⏐
D
1
Ψ
Base
−→
D
2
92
SHINICHI
MOCHIZUKI
[where
the
vertical
arrows
are
the
natural
projection
functors;
the
horizontal
arrows
are
equivalences
of
categories].
Moreover,
if
D
1
,
D
2
are
slim,
then
each
of
the
composite
functors
of
this
diagram
is
rigid.
(iii)
Suppose
further
that
C
1
,
C
2
are
of
rationally
standard
type.
Then
there
exists
an
isomorphism
of
functors
∼
Ψ
Φ
:
Φ
1
→
Φ
2
[where
we
regard,
for
i
=
1,
2,
the
functor
Φ
i
:
D
i
→
Mon
as
a
functor
on
D
i
]
lying
over
the
equivalence
of
categories
Ψ
Base
of
(i),
which
is
compatible
[when
the
C
i
are
of
isotropic,
but
not
of
group-like
type]
with
the
isomorphism
Ψ
Prime
of
Theorem
4.2,
(ii).
In
particular,
Ψ
Base
,
Ψ
Φ
induce
an
equivalence
of
categories
∼
Ψ
F
:
F
Φ
1
→
F
Φ
2
.
(iv)
Suppose
further
that
C
1
,
C
2
are
of
rationally
standard
type.
Then
there
exists
a
1-commutative
diagram
Ψ
C
1
⏐
⏐
−→
F
Φ
1
−→
Ψ
F
C
2
⏐
⏐
F
Φ
2
[where
the
vertical
arrows
are
the
functors
that
define
the
Frobenioid
structures
on
C
1
,
C
2
;
the
horizontal
arrows
are
equivalences
of
categories].
Moreover,
each
of
the
composite
functors
of
this
diagram
is
rigid.
Proof.
First,
we
observe
[cf.
Theorem
3.4,
(i)]
that
we
may
assume
without
loss
of
generality
that
C
1
,
C
2
are
of
isotropic
type
[cf.
Remark
4.5.1].
Also,
if
C
1
,
C
2
are
of
group-like
type
[cf.
Theorem
3.4,
(ii)],
then
“Div-slimness”
amounts
to
“slimness”,
so
assertions
(i),
(ii)
follow
from
Theorem
3.4,
(iv),
(v);
assertion
(iii)
is
vacuous;
assertion
(iv)
follows
from
the
fact
that
Ψ
preserves
Frobenius
degrees
[cf.
Theorem
3.4,
(iii),
(iv)].
Thus,
we
may
assume
without
loss
of
generality
that
C
1
,
C
2
are
not
of
group-like
type.
Now
we
consider
assertion
(i).
To
show
the
existence
and
1-uniqueness
of
a
1-commutative
diagram
as
in
the
statement
of
assertion
(i),
it
suffices
to
show
that
Ψ
preserves
“O
×
(−)”
[cf.
the
proof
of
Theorem
3.4,
(iv)].
But
observe
that,
for
A
∈
Ob(C
i
),
an
element
f
∈
O
×
(A)
determines
an
automorphism
[cf.
the
proof
of
Proposition
3.3,
(i)]
φ
f
∈
Aut((C
i
pl-bk
)
A
→
C
i
)
def
that
maps
to
the
identity
in
Aut((D
i
)
A
D
→
D
i
)
[where
A
D
=
Base(A)
—
cf.
∼
the
equivalence
of
categories
(C
i
pl-bk
)
A
→
(D
i
)
A
D
of
Definition
1.3,
(i),
(c)],
hence
also
to
the
identity
in
Aut((D
i
)
A
D
→
Mon)
[i.e.,
via
composition
with
Φ
i
].
Since
D
i
is
Div-slim,
it
thus
follows
that
the
elements
of
O
×
(A)
⊆
Aut
C
i
(A)
may
be
characterized
as
the
automorphisms
of
A
that
arise
from
automorphisms
φ
∈
Aut((C
i
pl-bk
)
A
→
C
i
)
THE
GEOMETRY
OF
FROBENIOIDS
I
93
such
that
every
automorphism
[of
an
object
of
C
i
]
induced
by
φ
is
a
Div-identity
automorphism.
Thus,
since
Ψ
preserves
pull-back
morphisms
[cf.
Theorem
3.4,
(iii)]
and
Div-identity
automorphisms
[cf.
Theorem
4.2,
(i);
our
assumption
that
the
Φ
i
are
perf-factorial],
we
thus
conclude
that
Ψ
preserves
“O
×
(−)”,
as
desired.
This
completes
the
proof
of
the
existence
and
1-uniqueness
of
a
1-commutative
diagram
as
in
the
statement
of
assertion
(i).
The
rigidity
assertion
in
the
statement
of
assertion
(i)
follows
by
observing
that
if
α
∈
Aut(C
i
→
C
i
un-tr
),
then
every
automorphism
[of
an
object
of
C
i
un-tr
]
induced
by
α
is
a
Div-identity
automorphism.
[Indeed,
this
follows
by
applying
the
functoriality
of
α
to
[co-angular]
pre-steps,
in
light
of
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d).]
In
particular,
it
follows
that
if
A
∈
Ob(C
i
),
def
A
D
=
Base(A),
then
the
element
∼
α
A
∈
Aut((C
i
pl-bk
)
A
→
D
i
)
→
Aut((D
i
)
A
D
→
D
i
)
determined
by
α
maps
[under
composition
with
Φ
i
:
D
i
→
Mon]
to
the
identity
element
of
Aut((D
i
)
A
D
→
Mon).
Thus,
since
D
i
is
Div-slim,
it
follows
that
every
automorphism
[of
an
object
of
C
i
un-tr
]
induced
by
α
is
a
base-identity
automor-
phism,
hence
trivial
[since
C
i
un-tr
is
of
unit-trivial
type].
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
First,
let
us
observe
that
we
obtain
a
1-
commutative
diagram
Ψ
−→
C
2
C
1
⏐
⏐
⏐
⏐
C
1
birat
Ψ
birat
−→
C
2
birat
[cf.
Corollary
4.10].
Since,
moreover,
the
base-isomorphisms
of
C
i
birat
are
precisely
the
morphisms
of
C
i
birat
which
are
abstractly
equivalent
to
morphisms
that
arise
from
base-isomorphisms
of
C
i
[cf.
Proposition
4.4,
(iv)],
it
follows
that
Ψ
birat
pre-
serves
base-isomorphisms,
hence
also
pull-back
morphisms
[cf.
Proposition
1.7,
(ii)].
Thus,
since
D
i
is
Div-slim,
the
base-identity
endomorphisms
of
A
∈
Ob(C
i
birat
)
may
be
characterized
as
the
endomorphisms
of
A
that
arise
from
endomorphisms
→
C
i
birat
)
φ
∈
End((C
i
birat
)
pl-bk
A
such
that
every
endomorphism
[of
an
object
of
C
i
birat
]
induced
by
φ
projects
to
an
automorphism
of
D
i
that
is
mapped
by
Φ
i
to
an
identity
automorphism.
Since,
by
Theorem
4.2,
(ii)
[cf.
also
the
fact
that
the
Φ
i
are
perf-factorial
and
non-
dilating],
it
follows
immediately
from
the
definition
of
C
i
birat
that
Ψ
birat
preserves
those
endomorphisms
[of
an
object
of
C
i
birat
]
that
project
to
an
automorphism
of
D
i
that
is
mapped
by
Φ
i
to
an
identity
automorphism,
we
thus
conclude
that
Ψ
birat
preserves
the
base-identity
endomorphisms
[hence,
in
particular,
that
Ψ
birat
preserves
“O
×
(−)”].
Thus,
we
obtain
a
1-commutative
diagram
C
1
birat
⏐
⏐
(C
1
birat
)
un-tr
Ψ
birat
−→
(Ψ
birat
)
un-tr
−→
C
2
birat
⏐
⏐
(C
2
birat
)
un-tr
94
SHINICHI
MOCHIZUKI
[where
the
vertical
arrows
are
the
natural
functors;
the
horizontal
arrows
are
equiv-
alences
of
categories].
Since,
moreover,
the
Frobenioids
(C
i
birat
)
un-tr
are
of
isotropic,
unit-trivial,
and
group-like
type,
we
thus
conclude
that
we
obtain
a
1-commutative
diagram
(Ψ
birat
)
un-tr
−→
(C
2
birat
)
un-tr
(C
1
birat
)
un-tr
⏐
⏐
⏐
⏐
D
1
Ψ
Base
−→
D
2
[cf.
Proposition
3.11,
(iii)].
Thus,
by
composing
diagrams,
we
obtain
a
1-commutative
diagram
as
in
the
statement
of
assertion
(ii),
which
is
easily
verified
to
be
1-unique.
Finally,
the
rigidity
assertion
in
the
statement
of
assertion
(ii)
follows
from
Propo-
sition
1.13,
(i).
This
completes
the
proof
of
assertion
(ii).
Next,
we
observe
that
assertion
(iii)
follows
formally
from
assertion
(ii);
Theo-
rem
4.9
[cf.
also
Definition
1.3,
(i),
(a),
(b);
the
technique
of
using
the
equivalence
∼
of
categories
“D
∗
→
D”
applied
in
Proposition
2.2,
(ii)].
Finally,
we
consider
asser-
tion
(iv).
In
light
of
the
structure
of
an
elementary
Frobenioid
[cf.
Definition
1.1,
(iii)],
the
existence
of
a
1-commutative
diagram
as
in
the
statement
of
assertion
(iv)
now
follows
simply
by
concatenating
assertions
(ii),
(iii),
with
the
fact
that
Ψ
preserves
Frobenius
degrees
[cf.
Theorem
3.4,
(iii)].
Finally,
the
rigidity
assertion
follows
via
the
same
argument
as
was
applied
to
prove
the
rigidity
assertion
that
appears
in
the
statement
of
assertion
(i).
This
completes
the
proof
of
assertion
(iv).
Remark
4.11.1.
Note
that
since
“slim
always
implies
Div-slim”,
it
follows
that,
at
least
when
the
divisorial
monoids
involved
are
perf-factorial,
Corollary
4.11,
(ii),
constitutes
a
substantial
strengthening
of
Theorem
3.4,
(v).
Remark
4.11.2.
Observe
that
in
Example
3.9,
since
the
subgroup
U
of
G
=
Aut(D
A
→
D)
[where
A
∈
Ob(D)]
acts
trivially
on
V
×
W
,
it
follows
that
D
fails
to
be
Div-slim,
so
the
non-preservation
of
units
that
occurs
in
this
example
does
not
contradict
Corollary
4.11,
(i),
(ii).
In
a
similar
vein,
in
Example
3.10,
since
G
=
Aut(D
A
→
D)
[where
A
∈
Ob(D)]
acts
trivially
on
Z
≥0
,
it
follows
that
D
fails
to
be
Div-slim,
so
the
non-preservation
of
base-identity
endomorphisms
that
occurs
in
this
example
does
not
contradict
Corollary
4.11,
(ii).
Corollary
4.12.
(Category-theoreticity
of
the
Functor
to
an
Elementary
Frobenioid
II)
For
i
=
1,
2,
let
Φ
i
be
a
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D
i
which
is
Frobenius-slim;
C
i
→
F
Φ
i
a
Frobenioid
of
rationally
standard
type;
0
D
i
the
monoid
on
D
i
that
assigns
to
every
object
of
D
i
the
monoid
with
one
element;
∼
Ψ
:
C
1
→
C
2
an
equivalence
of
categories.
If
C
1
,
C
2
are
of
group-like
type,
then
we
also
assume
that
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
base-isomorphisms.
Then
THE
GEOMETRY
OF
FROBENIOIDS
I
95
there
exists
a
1-unique
functor
Ψ
0
:
F
0
D
1
→
F
0
D
2
that
fits
into
a
1-commutative
diagram
Ψ
−→
C
2
C
1
⏐
⏐
⏐
⏐
F
0
D
1
Ψ
0
−→
F
0
D
2
[where
the
vertical
arrows
are
the
natural
projection
functors,
determined
by
the
Frobenius
degree
and
the
projection
to
D
i
[cf.
Proposition
4.4,
(i)];
the
horizontal
arrows
are
equivalences
of
categories].
Moreover,
if
D
1
,
D
2
are
slim,
then
each
of
the
composite
functors
of
this
diagram
is
rigid.
Proof.
First,
we
observe
[cf.
Theorem
3.4,
(i)]
that
we
may
assume
without
loss
of
generality
that
C
1
,
C
2
are
of
isotropic
type
[cf.
Remark
4.5.1].
Now
the
natural
projection
functors
C
i
→
F
0
D
i
may
be
identified
with
the
natural
functors
C
i
→
C
i
birat
→
(C
i
birat
)
un-tr
[cf.
Proposition
3.11,
(i)].
In
particular,
if
C
1
,
C
2
are
∼
of
group-like
type
[cf.
Theorem
3.4,
(ii)],
then
[since
C
i
→
C
i
birat
]
Corollary
4.12
follows
from
Theorem
3.4,
(iv).
Thus,
we
may
assume
without
loss
of
generality
that
C
1
,
C
2
are
not
of
group-like
type.
Then
Corollary
4.12
follows
by
applying
Corollary
4.10
to
pass
from
C
i
to
C
i
birat
[where
we
note
that,
by
Proposition
4.4,
(iv),
and
Theorem
3.4,
(iii),
it
follows
that
the
resulting
equivalence
of
categories
Ψ
birat
preserves
base-isomorphisms],
followed
by
Theorem
3.4,
(iv)
[where
we
note
that,
by
Proposition
4.8,
(iii),
C
i
birat
is
of
standard
type],
which
allows
us
to
pass
from
C
i
birat
to
(C
i
birat
)
un-tr
,
as
desired.
Finally,
the
rigidity
assertion
follows
from
Proposition
1.13,
(i).
This
completes
the
proof
of
Corollary
4.12.
Remark
4.12.1.
One
verifies
immediately
that
if
one
takes
the
group
G
of
Exam-
ple
3.10
to
be
residually
finite,
then
the
Frobenioid
of
Example
3.10
is
of
rationally
standard
and
unit-trivial
type
[but
not
of
group-like
type]
over
a
Frobenius-slim
base
category
[which
is
not
Div-slim
—
cf.
Remark
4.11.2].
In
particular,
one
may
apply
Corollary
4.12
to
the
self-equivalence
of
categories
of
Example
3.10.
On
the
other
hand,
since
this
self-equivalence
fails
to
preserve
base-identity
endomor-
phisms
of
Frobenius
type,
it
follows
that
it
is
not
possible
to
replace
the
“F
0
D
i
”
in
the
diagram
of
Corollary
4.12
by
“D
i
”.
96
SHINICHI
MOCHIZUKI
Section
5:
Model
Frobenioids
In
the
present
§5,
we
study
the
extent
to
which
an
arbitrary
Frobenioid
of
isotropic
type
may
be
constructed
explicitly
as
a
“model
Frobenioid”.
This
study
of
“model
Frobenioids”
will
be
of
use
in
the
consideration
of
the
concrete
examples
of
Frobenioids
that
we
discuss
in
§6
below.
In
the
following
discussion,
we
maintain
the
notation
of
§1,
§2,
§3,
§4.
In
particular,
we
assume
that
we
have
been
given
a
divisorial
monoid
Φ
on
a
connected,
totally
epimorphic
category
D
and
a
Frobenioid
C
→
F
Φ
.
Theorem
5.1.
(Divisorial
Descriptions)
Suppose
that
the
Frobenioid
C
is
def
of
isotropic
type.
Let
A,
A
∈
Ob(C)
be
Frobenius-trivial;
A
D
=
Base(A)
∈
def
Ob(D);
A
D
=
Base(A
)
∈
Ob(D);
D
isom
⊆
D
the
subcategory
determined
by
the
def
isom
isomorphisms
of
D;
D
D
=
(D
isom
)
D
[for
D
∈
Ob(D)
=
Ob(D
isom
)].
Write
def
Pic
Φ
(A)
=
Φ
gp
(A)/Φ
birat
(A)
[cf.
Proposition
4.4,
(iii)]
and
Pic
C
(A)
for
the
set
of
isomorphism
classes
of
C
×
D
isom
D
A
[where
the
fiber
product
category
is
taken
with
respect
to
the
natural
functors
D
isom
C
→
D,
D
A
→
D
—
cf.
§0].
Then:
D
(i)
The
assignment
that
maps
a
pair
of
pre-steps
(φ
:
B
→
A,
ψ
:
B
→
C)
to
the
object
isom
(C,
Base(φ)
◦
Base(ψ)
−1
)
∈
Ob(C
×
D
D
A
)
D
on
the
one
hand
and
to
the
element
Φ(φ)
−1
(Div(ψ)
−
Div(φ))
∈
Φ
gp
(A)
∼
on
the
other
hand
determines
a
bijection
Pic
Φ
(A)
→
Pic
C
(A).
Moreover,
if
(C,
ζ
:
∼
def
isom
C
D
→
A
D
)
∈
Ob(C
×
D
D
A
)
[where
C
∈
Ob(C),
C
D
=
Base(C)]
corresponds,
D
via
this
bijection,
to
an
element
γ
∈
Pic
Φ
(A),
and
κ
:
C
→
C
is
a
morphism
of
isom
Frobenius
type,
then
(C
,
ζ
◦
Base(κ)
−1
)
∈
Ob(C
×
D
D
A
)
corresponds
to
the
D
element
deg
Fr
(κ)
·
γ
∈
Pic
Φ
(A).
(ii)
If
∼
isom
);
(B,
λ
:
B
D
→
A
D
)
∈
Ob(C
×
D
D
A
D
def
∼
isom
(B
,
λ
:
B
D
→
A
D
)
∈
Ob(C
×
D
D
A
)
def
[where
B
D
=
Base(B);
B
D
=
Base(B
)],
then
there
exists
a
morphism
φ
:
B
→
B
D
THE
GEOMETRY
OF
FROBENIOIDS
I
97
in
C
of
Frobenius
degree
d
such
that
Base(φ)
=
(λ
)
−1
◦
θ
◦
λ,
where
θ
:
A
D
→
A
D
is
a
morphism
of
D,
and
Div(φ)
=
z
∈
Φ(B)
if
and
only
if
the
classes
β
∈
Pic
Φ
(A),
β
∈
Pic
Φ
(A
)
determined
by
B,
B
,
respectively,
via
the
bijection
of
(i)
satisfy
the
following
relation:
d
·
β
+
z|
A
D
=
(Φ(θ))(β
)
∈
Pic
Φ
(A)
[where,
by
abuse
of
notation,
we
denote
by
z|
A
D
the
image
of
Φ(λ)
−1
(z)
∈
Φ(A)
in
Pic
Φ
(A)].
Moreover,
if
such
a
morphism
exists,
then
its
unit-equivalence
class
[i.e.,
its
image
in
C
un-tr
,
or,
equivalently,
F
Φ
—
cf.
Proposition
3.3,
(iv)]
is
unique.
(iii)
The
subcategory
C
Fr-tr
⊆
C
determined
by
the
Frobenius-trivial
objects
and
isometric
morphisms
is
a
Frobenioid
of
isotropic,
group-like,
base-trivial,
and
Aut-ample
type.
In
particular,
the
isomorphism
class
of
a
Frobenius-trivial
object
of
C
is
completely
determined
by
the
isomorphism
class
of
its
projection
to
D;
all
Frobenius-trivial
objects
of
C
are
Aut-ample.
(iv)
Suppose
that
C
is
of
unit-trivial
type.
Then
any
skeletal
subcategory
P
⊆
(C
Fr-tr
)
pl-bk
determines
a
base-section
of
C;
any
base-section
of
C
admits
an
associated
Frobenius-section
F
.
Moreover,
C
is
of
model
type.
Proof.
First,
we
consider
assertion
(i).
Let
us
refer
to
a(n)
[ordered]
pair
of
pre-
steps
as
an
A-pair
if
the
first
pre-step
has
codomain
A,
and
the
second
pre-step
has
the
same
domain
as
the
first;
let
us
say
that
two
A-pairs
(φ
:
B
→
A,
ψ
:
B
→
C);
∼
(φ
:
B
→
A,
ψ
:
B
→
C
)
are
isomorphic
if
there
exist
isomorphisms
ι
B
:
B
→
B
,
∼
ι
C
:
C
→
C
such
that
φ
◦
ι
B
=
φ,
ψ
◦
ι
B
=
ι
C
◦
ψ.
Then
observe
that
by
the
equivalences
of
categories
of
Definition
1.3,
(iii),
(d),
it
follows
that
the
assignment
(φ
:
B
→
A,
ψ
:
B
→
C)
→
(Φ(φ)
−1
(Div(φ)),
Φ(φ)
−1
(Div(ψ)))
∈
Φ(A)
×
Φ(A)
determines
a
bijection
from
the
set
of
isomorphism
classes
of
A-pairs
onto
Φ(A)
×
Φ(A);
in
particular,
we
obtain
a
map
Φ(A)
×
Φ(A)
→
Pic
C
(A).
Moreover,
relative
to
this
bijection,
replacing
an
element
(x,
y)
∈
Φ(A)
×
Φ(A)
by
an
element
(x
+
z,
y
+
z)
∈
Φ(A)
×
Φ(A)
[where
z
∈
Φ(A)]
corresponds
to
replacing
(φ
:
B
→
A,
ψ
:
B
→
C)
by
(φ
◦
δ,
ψ
◦
δ),
for
some
pre-step
δ;
in
particular,
such
replacements
do
not
affect
the
element
of
Pic
C
(A)
determined
by
the
A-pair.
Now
I
claim
that
the
map
Φ(A)
×
Φ(A)
→
Pic
C
(A)
of
the
above
discussion
factors
through
Pic
Φ
(A).
Indeed,
suppose
that
(x,
y)
∈
Φ(A)
×
Φ(A),
(x
,
y
)
∈
Φ(A)
×
Φ(A)
map
to
the
same
element
of
Pic
Φ
(A).
Then,
by
the
definition
of
“Φ
birat
(A)”
[cf.
the
statements
and
proofs
of
Proposition
4.4,
(i),
(iii)],
it
follows
that
there
exists
a
pair
of
base-equivalent
pre-steps
δ
1
,
δ
2
:
D
→
A
such
that
Φ(δ
1
)
−1
(Div(δ
1
))
+
x
+
y
+
z
=
Φ(δ
2
)
−1
(Div(δ
2
))
+
x
+
y
+
z
for
some
z
∈
Φ(A)
[cf.
also
the
definition
of
“gp”
in
§0];
thus,
by
replacing
δ
1
,
δ
2
by
the
composite
of
δ
1
,
δ
2
with
an
appropriate
pre-step
[cf.
Definition
1.3,
(iii),
(d)],
we
may
assume
that
Φ(δ
1
)
−1
(Div(δ
1
))
=
x
+
y
+
z
;
Φ(δ
2
)
−1
(Div(δ
2
))
=
x
+
y
+
z
98
SHINICHI
MOCHIZUKI
for
some
z
∈
Φ(A)
[for
instance,
one
natural
choice
for
z
is
Φ(δ
1
)
−1
(Div(δ
1
))
+
x
+
y
+
z
=
Φ(δ
2
)
−1
(Div(δ
2
))
+
x
+
y
+
z];
by
replacing
(x,
y)
by
(x
+
z
,
y
+
z
)
[cf.
discussion
of
the
the
preceding
paragraph],
it
follows
that
we
may
assume,
without
loss
of
generality,
that
z
=
0.
Next,
by
applying
the
first
equivalence
of
categories
of
Definition
1.3,
(iii),
(d),
we
observe
that
there
exists
a
pre-step
δ
†
:
D
→
D
†
such
that
Div(δ
†
)
=
Φ(δ
i
)(x
+
x
+
y
+
y
),
where
i
=
1,
2
[and
we
note
that
Φ(δ
i
)
is
independent
of
i,
since
δ
1
,
δ
2
are
base-equivalent].
Thus,
[again
by
Definition
1.3,
(iii),
(d)]
we
conclude
that
there
exist
base-equivalent
pre-steps
δ
1
A
,
δ
2
A
:
A
→
D
†
such
that
δ
†
=
δ
2
A
◦
δ
1
=
δ
1
A
◦
δ
2
.
In
particular,
we
have
Div(δ
1
A
)
=
x
+
y
,
Div(δ
2
A
)
=
x
+
y.
Let
:
E
→
A
be
a
pre-step
with
Φ(
)
−1
(Div(
))
=
x
+
x
[cf.
Definition
1.3,
(iii),
(d)];
(φ
:
B
→
A,
ψ
:
B
→
C)
an
A-pair
that
corresponds
to
(x,
y);
(φ
:
B
→
A,
ψ
:
B
→
C
)
an
A-pair
that
corresponds
to
(x
,
y
).
Then
since
x,
x
≤
x
+
x
,
it
follows
[cf.
Definition
1.3,
(iii),
(d)]
that
there
exist
factorizations
=
φ
◦
η,
=
φ
◦
η
,
where
η
:
E
→
B,
η
:
E
→
B
are
pre-steps.
Moreover,
by
applying
the
the
second
equivalence
of
categories
of
Definition
1.3,
(iii),
(d),
to
D
†
,
we
conclude
from
the
existence
of
the
composites
of
:
E
→
A
with
δ
1
A
,
δ
2
A
:
A
→
D
†
that
there
exists
a
pre-step
F
:
F
→
D
†
and
a
pair
of
base-equivalent
pre-steps
δ
1
E
,
δ
2
E
:
E
→
F
such
that
the
following
relations
hold:
F
◦
δ
1
E
=
δ
1
A
◦
;
F
Div(δ
1
E
)
=
(Φ(
))(x
+
y
);
◦
δ
2
E
=
δ
2
A
◦
Div(δ
2
E
)
=
(Φ(
))(x
+
y)
[so
Φ(
F
)
−1
(Div(
F
))
=
Φ(δ
i
A
)
−1
(x
+
x
),
for
i
=
1,
2].
On
the
other
hand,
since
Div(ψ
◦
η)
=
(Φ(
))(x
+
y)
=
Div(δ
2
E
),
Div(ψ
◦
η
)
=
(Φ(
))(x
+
y
)
=
Div(δ
1
E
),
we
thus
conclude
[cf.
Definition
1.3,
(iii),
(d),
applied
to
the
pairs
of
pre-steps
(ψ
◦
η
:
E
→
C,
δ
2
E
:
E
→
F
)
and
(ψ
◦
η
:
E
→
C
,
δ
1
E
:
E
→
F
)
emanating
from
E]
∼
that
there
exists
an
isomorphism
ι
:
C
→
C
such
that
Base(ψ
◦η
)
=
Base(ι◦ψ
◦η),
Base(ι
◦
ψ)
◦
Base(φ)
−1
=
Base(ψ
)
◦
Base(φ
)
−1
.
That
is
to
say,
we
have
a
[not
necessarily
commutative!]
diagram
of
pre-steps
η
E
−→
⏐
⏐
η
φ
A
←−
B
φ
B
−→
A
⏐
⏐
ι◦ψ
ψ
−→
C
whose
projection
to
D
is
a
commutative
diagram
of
isomorphisms
that
is
compat-
ible
with
identification
of
the
two
copies
of
A
D
.
In
particular,
we
conclude
that
(C,
Base(φ)
◦
Base(ψ)
−1
),
(C
,
Base(φ
)
◦
Base(ψ
)
−1
)
determine
the
same
element
of
Pic
C
(A).
This
completes
the
proof
of
the
claim.
Thus,
we
obtain
a
map
Pic
Φ
(A)
→
Pic
C
(A).
It
follows
immediately
from
Definition
1.3,
(i),
(b),
that
this
map
is
a
surjection.
To
show
that
this
map
is
injective,
it
suffices
to
consider
(x,
y)
∈
Φ(A)
×
Φ(A),
(x
,
y
)
∈
Φ(A)
×
Φ(A)
that
map
to
the
same
element
of
Pic
C
(A).
Let
(φ
:
B
→
A,
ψ
:
B
→
C)
be
an
A-pair
that
corresponds
to
(x,
y);
(φ
:
B
→
A,
ψ
:
B
→
C)
an
A-pair
that
THE
GEOMETRY
OF
FROBENIOIDS
I
99
corresponds
to
(x
,
y
).
By
our
assumption
that
(x,
y)
and
(x
,
y
)
map
to
the
same
element
of
Pic
C
(A),
it
follows
that
we
may
assume
that
Base(φ
)
◦
Base(ψ
)
−1
=
Base(φ)
◦
Base(ψ)
−1
.
Thus,
by
applying
Definition
1.3,
(iii),
(d),
we
obtain
a
[not
necessarily
commutative!]
diagram
of
pre-steps
η
φ
E
−→
B
−→
A
⏐
⏐
⏐
⏐
ψ
η
A
φ
←−
B
ψ
−→
C
such
that
φ
◦
η
=
φ
◦
η
,
and
whose
projection
to
D
is
a
commutative
diagram
of
isomorphisms
that
is
compatible
with
identification
of
the
two
copies
of
A
D
.
Thus,
it
follows
that
ψ
◦
η,
ψ
◦
η
:
E
→
C
are
base-equivalent,
hence
determine
an
element
of
Φ
birat
(C),
which
may
be
transported
via
ψ,
φ
[or,
equivalently,
ψ
,
φ
]
to
an
element
of
Φ
birat
(A)
⊆
Φ
gp
(A)
which
[cf.
the
discussion
of
the
preceding
paragraph]
is
easily
verified
to
be
x
+y
−x
−y
∈
Φ
gp
(A).
This
completes
the
proof
of
the
injectivity,
hence
also
of
the
bijectivity
of
the
map
Pic
Φ
(A)
→
Pic
C
(A).
Also,
the
portion
of
assertion
(i)
concerning
morphisms
of
Frobenius
type
follows
easily
by
considering
commutative
diagrams
as
in
Proposition
1.10,
(i).
This
completes
the
proof
of
assertion
(i).
Now
assertion
(ii)
follows
formally
from
assertion
(i)
[cf.
also
Remark
1.1.1;
the
factorization
of
Definition
1.3,
(iv),
(a);
the
faithfulness
portion
of
Proposition
3.3,
(iv)].
Next,
we
consider
assertion
(iii).
First,
let
us
observe
that
by
assertion
(i),
any
∼
isom
isomorphism
α
D
:
A
D
→
A
D
determines
an
object
(A
,
α)
∈
Ob(C
×
D
D
A
)
which
D
[in
light
of
the
fact
that
A
is
Frobenius-trivial,
hence
admits
base-identity
endo-
morphisms
of
Frobenius
type
of
arbitrary
prescribed
Frobenius
degree]
corresponds
[via
the
bijection
of
assertion
(i)]
to
an
element
ξ
∈
Pic
Φ
(A)
such
that
d
·
ξ
=
ξ,
for
all
d
∈
N
≥1
.
Thus,
taking
d
=
2
implies
that
ξ
=
0,
i.e.,
[cf.
the
definition
of
∼
Pic
C
(A)]
that
there
exists
an
isomorphism
α
:
A
→
A
such
that
α
D
=
Base(α).
In
particular,
we
conclude
that
base-isomorphic
Frobenius-trivial
objects
of
C
are,
in
fact,
isomorphic,
and
that
all
Frobenius-trivial
objects
of
C
are
Aut-ample.
In
light
of
these
observations,
it
follows
immediately
that
C
Fr-tr
satisfies
the
conditions
of
Definition
1.3,
i.e.,
that
C
Fr-tr
is
a
Frobenioid
[of
isotropic,
group-like,
base-trivial,
and
Aut-ample
type].
This
completes
the
proof
of
assertion
(iii).
Finally,
we
consider
assertion
(iv).
First,
we
observe
that
since
C
is
of
unit-
trivial
type,
it
follows
immediately
[cf.,
e.g.,
Proposition
3.3,
(iii),
(iv)]
that
given
any
two
objects
A,
B
∈
Ob(C),
a
pull-back
morphism
A
→
B
(respectively,
base-
identity
endomorphism
of
Frobenius
type
of
A)
is
uniquely
determined
by
its
projec-
tion
to
D
(respectively,
by
its
Frobenius
degree).
Moreover,
by
assertion
(iii),
it
fol-
lows
immediately
that
if
A,
B
∈
Ob(C
Fr-tr
),
then
any
morphism
Base(A)
→
Base(B)
[in
D]
lifts
to
a
pull-back
morphism
of
C
Fr-tr
.
Thus,
we
conclude
that
the
natural
projection
functor
(C
Fr-tr
)
pl-bk
→
D
is
an
equivalence
of
categories,
hence
that
any
skeletal
subcategory
P
⊆
(C
Fr-tr
)
pl-bk
determines
a
base-section
of
C,
and
that
any
base-section
of
C
admits
an
associated
100
SHINICHI
MOCHIZUKI
Frobenius-section.
Moreover,
since
C
is
of
unit-trivial
type,
it
follows
immediately
from
the
structure
of
an
elementary
Frobenioid
[cf.
the
description
of
the
kernel
in
Proposition
4.4,
(iii)]
that
C
is
of
birationally
Frobenius-normalized
type,
hence
also
of
model
type,
as
desired.
This
completes
the
proof
of
assertion
(iv).
The
explicit
descriptions
of
Theorem
5.1,
(i),
(ii),
motivate
the
following
con-
struction/result.
Theorem
5.2.
(Model
Frobenioids)
Let
Φ
:
D
→
Mon
be
a
divisorial
monoid
on
D;
B
:
D
→
Mon
a
group-like
monoid
on
D;
Div
B
:
B
→
Φ
gp
a
homomorphism
of
monoids
on
D.
Denote
the
group-like
monoid
determined
by
the
image
of
Div
B
by
Φ
birat
⊆
Φ
gp
.
Then:
(i)
A
well-defined
category
C
may
be
constructed
in
the
following
fashion.
The
objects
of
C
are
pairs
of
the
form
(A
D
,
α)
def
def
def
where
A
D
∈
Ob(D),
α
∈
Φ(A
D
)
gp
;
set
Base(A)
=
A
D
,
Φ(A)
=
Φ(A
D
),
B(A)
=
B(A
D
).
A
morphism
def
def
φ
:
A
=
(A
D
,
α)
→
B
=
(B
D
,
β)
[where
A
D
,
B
D
∈
Ob(D),
α
∈
Φ(A)
gp
,
β
∈
Φ(B)
gp
]
is
defined
to
be
a
collection
of
data
as
follows:
(a)
an
element
deg
Fr
(φ)
∈
N
≥1
,
which
we
shall
refer
to
as
the
Frobenius
degree
of
φ;
(b)
a
morphism
Base(φ)
:
A
D
→
B
D
,
which
we
shall
refer
to
as
the
projection
to
D
to
φ;
(c)
an
element
Div(φ)
∈
Φ(A),
which
we
shall
refer
to
as
the
zero
divisor
of
φ;
(d)
an
element
u
φ
∈
B(A)
whose
image
Div
B
(u
φ
)
∈
Φ(A)
gp
satisfies
the
relation
deg
Fr
(φ)
·
α
+
Div(φ)
=
(Φ(Base(φ)))(β)
+
Div
B
(u
φ
)
in
Φ(A)
gp
.
The
composite
ψ
◦
φ
of
two
morphisms
φ
=
(deg
Fr
(φ),
Base(φ),
Div(φ),
u
φ
)
:
A
→
B
ψ
=
(deg
Fr
(ψ),
Base(ψ),
Div(ψ),
u
ψ
)
:
B
→
C
is
defined
as
follows:
ψ
◦
φ
=
deg
Fr
(ψ)
·
deg
Fr
(φ),
Base(ψ)
◦
Base(φ),
def
(Φ(Base(φ)))(Div(ψ))
+
deg
Fr
(ψ)
·
Div(φ),
(B(Base(φ)))(u
ψ
)
+
deg
Fr
(ψ)
·
u
φ
[cf.
Remark
1.1.1].
Moreover,
the
Frobenius
degree,
projection
to
D,
and
zero
divisor
determine
a
functor
C
→
F
Φ
.
THE
GEOMETRY
OF
FROBENIOIDS
I
101
(ii)
The
category
C
is
a
Frobenioid
[with
respect
to
the
functor
C
→
F
Φ
]
of
isotropic
and
model
—
hence,
in
particular,
birationally
Frobenius-normalized
—
type.
We
shall
refer
to
C
as
the
model
Frobenioid
defined
by
the
divisor
monoid
Φ
and
the
rational
function
monoid
B
[which
we
regard
as
equipped
with
the
homomorphism
Div
B
:
B
→
Φ
gp
].
Moreover,
there
is
a
natural
isomor-
phism
of
functors
between
the
functor
“O
×
(−)”
on
D
associated
to
the
Frobenioid
C
birat
[cf.
Propositions
2.2,
(ii),
(iii);
4.4,
(ii)]
and
the
functor
B;
this
isomorphism
is
compatible
with
the
homomorphisms
O
×
(−)
→
Φ
gp
[cf.
Proposition
4.4,
(iii)],
Div
B
:
B
→
Φ
gp
.
(iii)
C
is
of
standard
type
if
and
only
if
the
following
conditions
are
satisfied:
(a)
if
Φ
is
the
zero
monoid,
then
C
admits
a
Frobenius-compact
object;
(b)
D
is
of
FSMFF-type;
(c)
Φ
is
non-dilating.
C
is
of
rationally
standard
type
if
and
only
if
the
following
conditions
are
satisfied:
(a)
C
is
of
rational
and
standard
type;
(b)
(C
un-tr
)
birat
admits
a
Frobenius-compact
object.
(iv)
Suppose
that
Φ
=
Φ;
B
is
the
rational
function
monoid
on
D
associated
to
the
Frobenioid
C
[cf.
Proposition
4.4,
(ii)];
Div
B
:
B
→
Φ
gp
is
the
natural
homomorphism
O
×
(−)
→
Φ
gp
=
Φ
gp
[cf.
Proposition
4.4,
(iii)];
C
is
of
model
type.
Then
there
exists
an
equivalence
of
categories
∼
C
→C
that
is
1-compatible
with
the
functors
C
→
F
Φ
,
C
→
F
Φ
.
Proof.
Assertions
(i),
(ii)
follow
via
a
routine
verification
[which,
in
the
case
of
assertion
(ii),
is
reminiscent
of
the
verification
that
“elementary
Frobenioids
are
Frobenioids”
in
Proposition
1.5,
(i)];
in
light
of
assertion
(ii),
assertion
(iii)
follows
formally
from
the
definitions
[cf.
Definitions
3.1,
(i);
4.5,
(iii)].
Here,
we
observe
that
the
objects
A
=
(A
D
,
α)
such
that
α
=
0
are
Frobenius-trivial,
and
that
these
objects,
together
with
the
morphisms
φ
=
(deg
Fr
(φ),
Base(φ),
Div(φ),
u
φ
)
:
A
→
B
such
that
Div(φ)
=
0,
u
φ
=
1
[i.e.,
u
φ
is
the
identity
element
of
B(A)],
determine
a
base-Frobenius
pair
of
C.
Finally,
we
consider
assertion
(iv).
We
may
assume
without
loss
of
generality
that
C,
hence
also
C
Fr-tr
,
is
a
skeleton.
Let
(P,
F
)
be
a
base-Frobenius
pair
of
C
[cf.
our
assumption
that
C
is
of
model
type].
Thus,
P
may
be
regarded
as
a
subcategory
of
C
Fr-tr
.
If
C
∈
Ob(C),
then
let
us
refer
to
a(n)[ordered]
pair
of
pre-steps
in
C
(B
→
A,
A
→
C)
such
that
A
∈
Ob(P)
as
an
F
P-path
for
C.
Write
C
for
the
category
C
whose
objects
are
objects
of
C
equipped
with
an
F
P-path,
and
whose
morphisms
are
the
morphisms
between
the
objects
regarded
as
objects
of
C.
Thus,
we
have
a
natural
functor
C
→
C
[obtained
by
forgetting
the
F
P-paths],
102
SHINICHI
MOCHIZUKI
which
is
manifestly
an
equivalence
of
categories.
Thus,
it
suffices
to
construct
an
∼
equivalence
of
categories
C
→
C
that
is
compatible
with
the
functors
C
→
C
→
F
Φ
,
C
→
F
Φ
.
Next,
observe
that
we
may
apply
Remark
2.7.2
to
C
Fr-tr
[which
is
of
base-trivial
type,
by
Theorem
5.1,
(iii)]
to
conclude
that
every
morphism
φ
of
C
Fr-tr
admits
a
unique
factorization
φ
=
φ
P
◦
φ
O
×
◦
φ
F
in
C
Fr-tr
,
where
φ
P
is
P-distinguished;
φ
O
×
is
a
base-identity
automorphism;
φ
F
is
F
-distinguished.
Let
us
write
E
⊆
C
birat
for
the
full
subcategory
determined
by
the
image
of
the
objects
in
P.
Then
observe
∼
that
the
category
E
is
also
a
skeleton;
that
the
Frobenioid
E
→
C
birat
is
also
of
isotropic
and
base-trivial
type
[cf.
Proposition
4.8,
(i);
Theorem
5.1,
(iii)];
and
that
(P,
F
)
determine
a
base-Frobenius
pair
of
E.
Thus,
we
may
apply
Remark
2.7.2
to
E
to
conclude
that
every
morphism
ψ
of
E
admits
a
unique
factorization
ψ
=
ψ
P
◦
ψ
O
×
◦
ψ
F
in
E,
where
ψ
P
is
P-distinguished;
ψ
O
×
is
a
base-identity
automorphism;
ψ
F
is
F
-distinguished.
Now
observe
that
to
every
object
C
∈
Ob(C)
equipped
with
an
F
P-path
(ζ
A
:
B
→
A,
ζ
C
:
B
→
C),
we
may
associate
an
object
(Base(A),
Φ(ζ
A
)
−1
(Div(ζ
C
)
−
Div(ζ
A
))
∈
Φ
gp
(A))
of
C
[cf.
Theorem
5.1,
(i)].
If
C
∈
Ob(C)
is
equipped
with
an
F
P-path
(ζ
A
:
B
→
A
,
ζ
C
:
B
→
C
),
then
we
may
associate
to
any
morphism
φ
:
C
→
C
a
morphism
deg
Fr
(φ),
Base(ζ
A
)
◦
Base(ζ
C
)
−1
◦
Base(φ)
◦
Base(ζ
C
)
◦
Base(ζ
A
)
−1
:
A
→
A
,
(Φ(ζ
A
)
−1
◦
Φ(ζ
C
))(Div(φ))
∈
Φ(A),
birat
birat
−1
birat
birat
−1
{ζ
A
◦
(ζ
C
)
◦
φ
birat
◦
ζ
C
◦
(ζ
A
)
}
O
×
∈
O
×
(A
birat
)
[where
the
superscript
“birat’s”
denote
the
images
of
the
respective
objects
and
morphisms
of
C
in
C
birat
]
of
C.
Now
in
light
of
the
fact
that
C
is
of
model
—
hence,
in
particular,
birationally
Frobenius-normalized
—
type,
it
is
a
routine
exercise
to
verify
that
these
assignments
determine
a
functor
C
→
C
that
is
compatible
with
the
functors
C
→
C
→
F
Φ
,
C
→
F
Φ
.
Indeed,
this
is
immediate
for
the
first
three
entries
of
the
data
that
define
a
morphism
of
C;
for
the
final
entry,
it
follows
from
the
existence
of
the
unique
factorizations
of
morphisms
of
E
discussed
above.
Note
that
these
factorizations
also
imply
that
this
functor
THE
GEOMETRY
OF
FROBENIOIDS
I
103
C
→
C
is
faithful.
Moreover,
this
functor
C
→
C
is
manifestly
essentially
surjective
[cf.
Theorem
5.1,
(i)]
and
full
[cf.
Theorem
5.1,
(ii)],
hence
an
equivalence
of
categories,
as
desired.
This
completes
the
proof
of
assertion
(iv).
Remark
5.2.1.
It
follows
formally
from
Theorem
5.2,
(ii),
(iv),
that
the
Frobe-
nioid
“C”
of
Example
4.6
constitutes
an
example
of
a
Frobenioid
of
isotropic,
stan-
dard,
and
[strictly]
rational
type,
which
is
not
of
group-like
or
model
type.
Proposition
5.3.
(Realifications
of
Frobenioids)
Suppose
that
Φ
is
perf-
factorial.
Then
we
shall
refer
to
as
the
realification
C
rlf
of
the
Frobenioid
C
the
model
Frobenioid
[cf.
Theorem
5.2,
(ii)]
associated
to
the
divisor
monoid
Φ
rlf
[i.e.,
the
“realification”
of
Definition
2.4,
(i)]
and
the
rational
function
monoid
R
·
Φ
birat
⊆
(Φ
rlf
)
gp
[i.e.,
for
A
D
∈
Ob(D),
(R
·
Φ
birat
)(A
D
)
is
the
R-vector
subspace
of
(Φ
rlf
)
gp
(A
D
)
generated
by
Φ
birat
(A
D
)].
Moreover,
the
Frobenioid
C
un-tr
(respec-
tively,
(C
un-tr
)
pf
)
is
of
model
type
and
may
be
obtained
as
the
model
Frobenioid
associated
to
the
divisor
monoid
Φ
(respectively,
Φ
pf
)
and
the
rational
function
monoid
Φ
birat
(respectively,
Q
·
Φ
birat
=
Φ
birat
⊗
Z
Q
=
(Φ
birat
)
pf
).
In
particular,
if
C
is
of
Frobenius-isotropic
type,
then
there
is
a
natural
1-commutative
diagram
of
functors
C
−→
C
istr
⏐
⏐
−→
C
pf
⏐
⏐
C
un-tr
−→
(C
un-tr
)
pf
−→
C
rlf
[where
the
functor
C
→
C
istr
is
the
isotropification
functor
of
Proposition
1.9,
(v);
the
remaining
functors
are
the
functors
that
arise
naturally
from
the
construction
of
the
“unit-trivialization”,
“perfection”,
and
“realification”].
Proof.
Since
Frobenioids
of
unit-trivial
type
are
always
of
model
type
[cf.
The-
orem
5.1,
(iv)],
the
various
assertions
in
the
statement
of
Proposition
5.3
follow
immediately
from
the
definitions
and
Theorem
5.2,
(ii),
(iv).
Corollary
5.4.
(Category-theoreticity
of
the
Realification)
For
i
=
1,
2,
let
Φ
i
be
a
perf-factorial
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D
i
which
is
Div-slim
[with
respect
to
Φ
i
];
C
i
→
F
Φ
i
a
Frobenioid
of
rationally
standard
type;
∼
Ψ
:
C
1
→
C
2
an
equivalence
of
categories.
If
C
1
,
C
2
are
of
group-like
type,
then
we
also
assume
that
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
base-isomorphisms.
Then
104
SHINICHI
MOCHIZUKI
there
exists
a
1-unique
functor
Ψ
rlf
:
C
1
rlf
→
C
2
rlf
that
fits
into
a
1-commutative
diagram
Ψ
C
1
−→
C
2
⏐
⏐
⏐
⏐
C
1
rlf
Ψ
rlf
−→
C
2
rlf
[where
the
vertical
arrows
are
the
natural
functors
of
Proposition
5.3;
the
horizontal
arrows
are
equivalences
of
categories].
Moreover,
each
of
the
composite
functors
of
this
diagram
is
rigid.
Finally,
the
formation
of
Ψ
rlf
from
Ψ
is
1-compatible
with
the
1-commutative
diagram
of
Proposition
5.3
[involving
perfections,
unit-
trivializations,
etc.].
Proof.
In
light
of
the
definition
of
the
realification
[cf.
Proposition
5.3],
Corollary
5.4
follows
immediately
from
Corollaries
4.10;
4.11,
(iii),
(iv).
[Here,
we
note
that
the
rigidity
assertion
of
Corollary
5.4
follows
by
a
similar
argument
applied
to
prove
the
rigidity
assertion
in
Corollary
4.11,
(i),
(iv).]
Before
continuing,
we
note
the
following
[portions
of
which
were
in
fact
applied
in
the
proofs
of
Theorems
4.2,
4.9].
Proposition
5.5.
(Perfection,
Unit-trivialization
and
Realification
of
Types)
Suppose
that
C
is
of
Frobenius-isotropic
and
Frobenius-normalized
type.
Then:
(i)
If
A
∈
Ob(C
istr
)
maps
to
an
object
A
pf
∈
Ob(C
pf
),
then
the
natural
functor
∼
C
→
C
pf
determines
a
natural
isomorphism
O
(A)
pf
→
O
(A
pf
).
(ii)
There
is
a
natural
equivalence
of
categories
[compatible
with
the
func-
tors
to
the
respective
elementary
Frobenioids]
between
(C
pf
)
un-tr
and
(C
un-tr
)
pf
and
between
(C
pf
)
birat
and
(C
birat
)
pf
.
(iii)
If
C
is
of
standard
(respectively,
rationally
standard;
model)
type,
then
so
is
C
pf
.
Moreover,
C
un-tr
,
C
rlf
are
always
of
model
type.
Finally,
suppose
further
that
C
is
not
of
group-like
type.
Then
if
C
is
of
standard
(respectively,
rationally
standard)
type,
then
so
are
C
un-tr
,
C
rlf
.
(iv)
If
C
is
the
model
Frobenioid
associated
to
data
Φ,
B,
Div
B
:
B
→
Φ
gp
[cf.
Theorem
5.2,
(ii)],
then
there
is
a
natural
equivalence
of
categories
[compatible
with
the
functors
to
the
respective
elementary
Frobenioids]
between
C
pf
(respectively,
C
un-tr
;
C
rlf
)
and
the
model
Frobenioid
associated
to
the
data
Φ
pf
,
B
pf
,
B
pf
→
(Φ
gp
)
pf
(respectively,
Φ,
Φ
birat
,
Φ
birat
→
Φ
gp
;
Φ
rlf
,
R
·
Φ
birat
,
R
·
Φ
birat
→
(Φ
rlf
)
gp
).
Proof.
Assertion
(i)
follows
immediately
for
Frobenius-trivial
A
by
considering
base-identity
endomorphisms
of
Frobenius
type
of
A
and
applying
the
hypothesis
that
C
is
of
Frobenius-normalized
type;
the
case
of
arbitrary
A
then
follows
by
con-
sidering
“pairs
of
pre-steps”
as
in
Theorem
5.1,
(i)
[cf.
also
Definition
1.3,
(iii),
THE
GEOMETRY
OF
FROBENIOIDS
I
105
(c)].
Next,
we
consider
assertion
(ii).
One
checks
immediately
that
[in
light
of
our
hypothesis
that
C
is
of
Frobenius-isotropic
type]
we
may
assume
without
loss
of
gen-
erality
that
C
is
of
isotropic
type.
Then
it
follows
immediately
from
the
definition
of
the
perfection
[cf.
Definition
3.1,
(iii)]
that
it
suffices
to
obtain
natural
bijections
be-
tween
the
respective
sets
of
morphisms
between
the
images
of
two
given
objects
of
C
in
(C
pf
)
un-tr
,
(C
un-tr
)
pf
(respectively,
(C
pf
)
birat
),
(C
birat
)
pf
).
But
this
follows
immedi-
ately
from
the
definitions,
together
with
Proposition
3.2,
(ii),
applied
to
“pre-steps”
and
“units”
[i.e.,
base-identity
automorphisms].
Next,
we
consider
assertion
(iii).
First,
we
observe
that
C
un-tr
,
C
rlf
are
of
model
type
[cf.
Theorem
5.1,
(iv);
Proposi-
tion
5.3;
Theorem
5.2,
(ii)],
hence
of
isotropic
and
birationally
Frobenius-normalized
type
[cf.
Definitions
2.7,
(iii);
4.5,
(i)].
Next,
let
us
observe
that
by
assertion
(ii),
we
∼
∼
have
natural
equivalences
((C
un-tr
)
birat
)
pf
→
((C
pf
)
un-tr
)
birat
,
(C
birat
)
pf
→
(C
pf
)
birat
;
moreover,
since
C
un-tr
is
of
birationally
Frobenius-normalized
type,
it
follows
that
(C
un-tr
)
birat
is
of
Frobenius-normalized
type,
so
assertion
(i)
may
be
applied
to
(C
un-tr
)
birat
.
In
light
of
these
observations,
assertion
(iii)
for
C
pf
follows
immedi-
ately
from
the
definitions
[cf.
also
Proposition
3.2,
(ii),
(iii)]
by
observing
that
C
pf
is
of
isotropic
type,
and
that
by
assertion
(i),
if
C
∗
is
C
or
(C
un-tr
)
birat
[or
C
birat
,
when
C
is
of
birationally
Frobenius-normalized
type],
and
A
∈
Ob((C
∗
)
istr
),
then
O
(−)
of
the
image
of
A
in
(C
∗
)
pf
is
the
perfection
of
O
(A).
Now
suppose
that
C,
hence
also
C
un-tr
,
C
rlf
,
are
not
of
group-like
type.
Since
(C
un-tr
)
birat
admits
a
Frobenius-
compact
object,
the
same
is
true
for
(C
rlf
)
birat
.
Also,
we
observe
that
the
pull-back
morphisms
of
C
un-tr
,
C
rlf
are
precisely
the
linear
isometries
[cf.
Proposition
1.4,
(ii)].
In
light
of
these
observations,
it
follows
immediately
from
the
definitions
that
if
C
is
of
standard
(respectively,
rationally
standard)
type,
then
so
are
C
un-tr
,
C
rlf
.
Finally,
assertion
(iv)
is
immediate
from
the
definitions
[cf.
also
assertions
(i),
(ii);
Proposition
5.3].
Finally,
we
conclude
the
theory
of
the
present
§5
by
discussing
a
certain
issue
which
is
closely
related
to
the
issue
of
being
of
model
type.
Namely,
instead
of
working
at
the
level
of
the
entire
category
C,
or
C
Fr-tr
,
we
consider
the
issue
of
being
“of
model
type”
at
the
level
of
a
single
Frobenius-trivial
object:
Proposition
5.6.
(Base-Sections
of
Frobenius-Trivial
Objects)
Suppose
that
C
is
of
model
[hence,
in
particular,
isotropic
—
cf.
Definition
2.7,
(iii)]
and
unit-profinite
type.
Let
(P,
F
)
be
a
base-Frobenius
pair
of
C;
A
∈
Ob(P)
a
def
Frobenius-trivial
object;
A
D
=
Base(A).
Then
the
pair
σ
:
Aut
D
(A
D
)
→
Aut
C
(A),
φ
:
N
≥1
→
End
C
(A)
—
where
σ
is
a
group
homomorphism
whose
composite
with
the
natural
surjec-
tion
Aut
C
(A)
Aut
D
(A
D
)
[cf.
Theorem
5.1,
(iii)]
is
the
identity,
and
φ
is
a
homomorphism
of
monoids
—
determined
by
“restricting”
P,
F
to
A,
in
fact,
de-
pends
only
on
the
data
(C,
A),
and,
in
particular,
is
independent
of
the
data
(F
,
P)
—
up
to
conjugation
[as
a
pair!]
by
an
element
of
O
×
(A).
We
shall
refer
to
such
a
pair
(σ,
φ)
as
a
base-Frobenius
pair
of
A;
when
F
is
regarded
as
being
106
SHINICHI
MOCHIZUKI
known
only
up
to
composition
with
automorphisms
of
the
monoid
N
≥1
,
we
shall
refer
to
such
a
pair
as
a
quasi-base-Frobenius
pair
of
A.
Proof.
Let
σ
:
Aut
D
(A
D
)
→
Aut
C
(A),
φ
:
N
≥1
→
End
C
(A)
be
another
such
pair,
that
arises
from
a
base-Frobenius
pair
(P
,
F
)
of
C.
Write
def
def
E
⊆
End
C
(A)
for
the
submonoid
of
base-isomorphisms;
φ
n
=
φ(n)
∈
E,
φ
n
=
φ
(n)
∈
E,
for
n
∈
N
≥1
.
Then
I
claim
that
it
suffices
to
show
the
existence
of
a
u
∈
O
×
(A)
⊆
E
such
that
u
·
φ
p
·
u
−1
=
φ
p
def
def
for
all
p
∈
Primes.
Indeed,
if,
for
α
∈
Aut
D
(A
D
),
we
write
σ
α
=
σ(α),
σ
α
=
σ
(α)
—
so
σ
α
=
v
α
·
u
·
σ
α
·
u
−1
,
for
some
v
α
∈
O
×
(A)
⊆
E
—
then
it
follows
from
the
functoriality
of
F
,
F
that,
for
p
∈
Primes,
σ
α
·
φ
p
=
φ
p
·
σ
α
;
σ
α
·
φ
p
=
φ
p
·
σ
α
—
hence
[since
C,
being
of
model
type,
is
also
of
[birationally]
Frobenius-normalized
type
—
cf.
Definition
4.5,
(i)]
that
u
·
v
α
·φ
p
·
σ
α
·
u
−1
=
v
α
·
u
·
σ
α
·
φ
p
·
u
−1
=
v
α
·
(u
·
σ
α
·
φ
p
·
u
−1
)
=
v
α
·
(u
·
σ
α
·
u
−1
)
·
(u
·
φ
p
·
u
−1
)
=
σ
α
·
φ
p
=
φ
p
·
σ
α
=
(u
·
φ
p
·
u
−1
)
·
v
α
·
(u
·
σ
α
·
u
−1
)
=
(u
·
v
α
p
·
φ
p
·
u
−1
)
·
(u
·
σ
α
·
u
−1
)
=
u
·
v
α
p
·
φ
p
·
σ
α
·
u
−1
—
which
[by
the
total
epimorphicity
of
C]
implies
that
v
α
=
v
α
p
,
for
all
p
∈
Primes.
Thus,
by
taking
p
=
2,
we
obtain
that
v
α
=
1.
Since
φ,
φ
are
homomorphisms,
and
N
≥1
is
generated
by
Primes,
this
completes
the
proof
of
the
claim.
To
verify
the
existence
of
a
u
∈
O
×
(A)
as
in
the
above
claim,
let
us
first
observe
that
if
M
⊆
O
×
(A)
⊆
E
is
any
subgroup
such
that
for
any
m
∈
M
,
f
∈
E,
there
exists
an
m
∈
M
such
that
f
·
m
=
m
·
f
,
then
there
is
a
natural
monoid
def
structure
on
the
set
of
cosets
E
M
=
M
\E
=
{M
·
f
}
f
∈E
,
together
with
a
natural
surjection
of
monoids
E
E
M
.
For
p
∈
Primes,
let
us
write
M
p
⊆
O
×
(A)
for
the
closed
subgroup
topologically
generated
by
the
pro-l
portions
(O
×
(A))[l]
[cf.
Definition
2.8,
(ii)]
of
O
×
(A),
as
l
ranges
over
the
primes
=
p.
Note
that
since
the
Frobenioid
C
Fr-tr
is
of
Aut-ample
type
[cf.
Theorem
5.1,
(iii)],
it
follows
that
any
f
∈
E
admits
a
factorization
f
=
f
0
·
f
1
,
where
f
0
is
an
automorphism,
and
f
1
is
a
base-identity
endomorphism.
Thus,
[by
applying,
again,
the
fact
that
C,
being
of
model
type,
is
also
of
[birationally]
Frobenius-normalized
type
—
cf.
Definition
4.5,
(i)]
it
follows
that
“for
any
m
∈
M
p
,
there
exists
an
m
∈
M
p
such
THE
GEOMETRY
OF
FROBENIOIDS
I
107
that
f
·
m
=
m
·
f
”.
In
particular,
it
makes
sense
to
speak
of
the
monoid
E
M
p
.
Let
p
us
use
the
symbol
“
≈
”
to
denote
the
equality
of
the
images
in
E
p
of
elements
of
E.
Now
since
we
have
a
natural
isomorphism
∼
O
×
(A)[p]
→
O
×
(A)
p∈Primes
[cf.
Definition
2.8,
(ii)],
it
thus
follows
that
to
prove
the
existence
of
a
u
∈
O
×
(A)
as
desired,
it
suffices
to
show,
for
each
p
∈
Primes,
the
existence
of
a
u
p
∈
O
×
(A)[p]
p
≈
φ
l
,
for
all
l
∈
Primes
[i.e.,
we
then
take
u
to
be
the
“infinite
such
that
u
p
·φ
l
·u
−1
p
product”
of
the
u
p
].
p
Now
observe
that
for
each
l
∈
Primes,
φ
l
≈
v
l
·
φ
l
,
for
some
v
l
∈
O
×
(A)[p].
p
Since,
for
w
∈
O
×
(A)[p],
we
have,
for
l
∈
Primes,
w
·
φ
l
·
w
−1
≈
w
1−l
·
φ
l
[where
we
recall
again
that
C,
being
of
model
type,
is
also
of
[birationally]
Frobenius-normalized
type
—
cf.
Definition
4.5,
(i)],
and
O
×
(A)[p]
is
a
[topologically
finitely
generated]
p
≈
φ
p
,
pro-p
group,
it
follows
that
there
exists
a
u
p
∈
O
×
(A)[p]
such
that
u
p
·φ
p
·u
−1
p
p
as
well
as
a
w
l
∈
O
×
(A)[p]
such
that
w
l
·
u
p
·
φ
l
·
u
−1
≈
φ
l
,
for
each
l
∈
Primes.
p
On
the
other
hand,
since
φ,
φ
are
homomorphisms,
it
follows
that
p
φ
l
1
·
φ
l
2
≈
φ
l
2
·
φ
l
1
;
p
φ
l
1
·
φ
l
2
≈
φ
l
2
·
φ
l
1
[for
l
1
,
l
2
∈
Primes].
Thus,
for
l
∈
Primes,
we
have
p
p
−1
≈
w
l
·
u
p
·
φ
l
·
φ
p
·
u
−1
≈
w
l
·
u
p
·
φ
l
·
u
−1
w
l
·
u
p
·φ
p
·
φ
l
·
u
−1
p
p
p
·
u
p
·
φ
p
·
u
p
p
p
p
−1
≈
φ
l
·
φ
p
≈
φ
p
·
φ
l
≈
u
p
·
φ
p
·
u
−1
p
·
w
l
·
u
p
·
φ
l
·
u
p
p
p
−1
≈
u
p
·
w
l
p
·
φ
p
·
u
−1
≈
w
l
p
·
u
p
·
φ
p
·
φ
l
·
u
−1
p
·
u
p
·
φ
l
·
u
p
p
p
—
which
[by
the
total
epimorphicity
of
C]
implies
that
w
l
≈
w
l
p
[for
all
l
∈
Primes].
Since
O
×
(A)[p]
is
a
[topologically
finitely
generated]
pro-p
group,
we
thus
conclude
p
that
w
l
≈
1.
This
completes
the
proof
of
the
existence
of
a
u
∈
O
×
(A)
as
desired,
and
hence
of
Proposition
5.6.
Remark
5.6.1.
The
notion
of
a
“base-section
of
a
Frobenius-trivial
object”
[i.e.,
in
the
notation
of
Proposition
5.6,
a
section
“σ”]
is
intended
to
be
an
abstract
category-theoretic
translation
of
the
notion
of
a
“tautological
section
of
a
trivial
line
bundle”
[cf.
Remark
2.7.1;
the
Frobenioids
of
Examples
6.1,
6.3
below].
Corollary
5.7.
(Category-theoreticity
of
Base-Sections)
For
i
=
1,
2,
let
Φ
i
be
a
perf-factorial
divisorial
monoid
on
a
connected,
totally
epimorphic
category
D
i
which
is
Div-slim
[with
respect
to
Φ
i
];
C
i
→
F
Φ
i
a
Frobenioid
of
standard
type;
∼
Ψ
:
C
1
→
C
2
108
SHINICHI
MOCHIZUKI
an
equivalence
of
categories.
If
C
1
,
C
2
are
of
group-like
type,
then
we
also
assume
that
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
base-isomorphisms.
Then:
(i)
Ψ
maps
base-sections
(respectively,
quasi-base-Frobenius
pairs)
of
C
1
to
base-sections
(respectively,
quasi-base-Frobenius
pairs)
of
C
2
.
In
particular,
C
1
is
of
model
type
if
and
only
if
C
2
is.
(ii)
C
1
is
of
unit-profinite
type
if
and
only
if
C
2
is.
(iii)
Suppose
that
C
1
,
C
2
are
of
model
and
unit-profinite
type.
Then
Ψ
maps
every
quasi-base-Frobenius
pair
of
a
Frobenius-trivial
object
A
1
∈
Ob(C
1
)
to
a
quasi-base-Frobenius
pair
of
a
Frobenius-trivial
object
A
2
∈
Ob(C
2
).
(iv)
Suppose,
moreover,
when
C
1
,
C
2
are
of
group-like
type,
that
both
Ψ
and
some
quasi-inverse
to
Ψ
preserve
Frobenius
degrees.
Then
the
prefix
“quasi-”
may
be
removed
from
the
statements
of
(i),
(iii).
Proof.
Indeed,
sorting
through
the
definitions,
to
verify
assertions
(i),
(ii),
(iii),
(iv)
it
suffices
to
show
that
Ψ
preserves
isotropic
objects,
prime-Frobenius
morphisms,
pull-back
morphisms,
birationalizations,
the
natural
projection
functor
C
i
→
D
i
[hence,
in
particular,
the
units
“O
×
(−)”],
and
[in
the
case
of
the
final
portion
of
assertion
(iv),
when
C
1
,
C
2
are
not
of
group-like
type]
Frobenius
degrees.
But
this
follows
from
Theorem
3.4,
(i),
(iii);
Corollary
4.10;
Corollary
4.11,
(ii)
[cf.
also
Remark
3.4.1].
This
completes
the
proof
of
Corollary
5.7.
THE
GEOMETRY
OF
FROBENIOIDS
I
109
Section
6:
Some
Motivating
Examples
In
the
present
§6,
we
discuss
some
of
the
principal
motivating
examples
from
arithmetic
geometry
of
the
theory
of
Frobenioids.
In
particular,
in
the
case
of
number
fields,
one
of
these
examples
provides
an
interesting
“category-theoretic
interpretation”
of
some
results
of
classical
number
theory,
such
as
the
Dirichlet
unit
theorem
and
Tchebotarev’s
density
theorem,
as
well
as
a
result
in
transcendence
theory
due
to
Lang
[cf.
Theorem
6.4,
(i),
(iii),
(iv)].
Example
6.1.
A
Frobenioid
of
Geometric
Origin.
Let
V
be
a
proper
normal
[geometrically
integral]
variety
over
a
field
k;
K
the
function
field
of
V
;
def
D
K
a
set
of
Q-Cartier
K/K
a
[possibly
infinite]
Galois
extension;
G
=
Gal(
K/K);
prime
divisors
on
V
.
The
connected
objects
of
the
Galois
category
B(G)
[cf.
§0]
may
is
a
finite
[necessarily
separable]
be
thought
of
as
schemes
Spec(L),
where
L
⊆
K
extension
of
K.
If
we
write
V
[L]
for
the
normalization
of
V
in
L
[so
V
[L]
is
also
a
proper
normal
variety],
then
let
us
write
D
L
for
the
set
of
prime
divisors
of
V
[L]
that
map
into
[possibly
subvarieties
of
codimension
≥
1
of]
prime
divisors
of
D
K
.
If
for
every
Spec(L)
∈
Ob(B(G)
0
)
[cf.
§0],
every
prime
divisor
of
D
L
is
Q-Cartier,
In
the
following,
we
shall
assume
that
then
we
shall
say
that
D
K
is
K-Q-Cartier.
D
K
is
K-Q-Cartier.
If
L
⊆
K
is
a
finite
extension,
then
let
us
write
Φ(L)
⊆
Z
≥0
[D
L
]
⊆
Z[D
L
]
for
the
monoid
of
Cartier
effective
divisors
D
on
V
[L]
with
support
in
D
L
[i.e.,
D
such
that
every
prime
divisor
in
the
support
of
D
belongs
to
D
L
]
and
B(L)
⊆
L
×
for
the
group
of
rational
functions
f
on
V
[L]
such
that
every
prime
divisor
at
which
f
has
a
zero
or
a
pole
belongs
to
D
L
.
Observe
that
Φ(L)
gp
⊆
Z[D
L
]
may
be
identified
with
the
group
of
Cartier
divisors
on
V
[L],
and
that
Φ(L)
pf
=
Q
≥0
[D
L
]
⊆
Q[D
L
]
=
(Φ(L)
pf
)
gp
[since
D
K
is
K-Q-Cartier];
moreover,
one
verifies
immediately
that
Φ(L)
is
perf-
∼
factorial,
that
there
is
a
natural
bijection
Prime(Φ(L))
→
D
L
,
and
that
the
supports
of
elements
of
Φ(L)
are
precisely
the
finite
subsets
of
D
L
.
By
assigning
to
a
rational
function
f
the
divisor
obtained
by
subtracting
the
divisor
of
poles
of
f
from
the
divisor
of
zeroes
of
f
,
we
obtain
a
natural
homomorphism
B(L)
→
Φ(L)
gp
which
is
functorial
in
L.
In
particular,
the
assignments
L
→
Φ(L),
L
→
B(L)
def
determine,
respectively,
a
perf-factorial
divisorial
monoid
Φ
on
D
=
B(G)
0
and
a
group-like
monoid
B
on
D,
equipped
with
a
homomorphism
[of
monoids
on
D]
B
→
Φ
gp
.
Thus,
by
Theorem
5.2,
(ii),
this
data
determines
a
[model]
Frobenioid
C
V,
K,D
K
110
SHINICHI
MOCHIZUKI
of
isotropic
and
birationally
Frobenius-normalized
type.
Note
that
an
object
of
C
V,
K,D
K
that
projects
to
Spec(L)
∈
Ob(B(G)
0
)
may
be
thought
of
as
a
line
bundle
L
on
V
[L]
that
is
representable
by
a
Cartier
divisor
D
with
support
in
D
L
.
If
L
is
such
a
line
bundle
on
V
[L],
and
M
is
such
a
line
bundle
on
V
[M
]
[where
M
⊆
K
is
a
finite
extension
of
K],
then
one
verifies
immediately
that
a
morphism
L
→
M
in
C
V,
K,D
K
may
be
thought
of
as
consisting
of
the
following
data:
(a)
a
morphism
Spec(L)
→
Spec(M
)
over
Spec(K)
[which
thus
induces
a
morphism
V
[L]
→
V
[M
]
over
V
];
(b)
an
element
d
∈
N
≥1
;
(c)
a
morphism
of
line
bundles
L
⊗d
→
M|
V
[L]
on
V
[L]
whose
zero
locus
is
a
Cartier
divisor
supported
in
D
L
.
Also,
we
observe
that
[since
V
[L]
is
a
proper
normal
variety]
for
A
∈
Ob(C
V,
K,D
K
)
that
projects
to
Spec(L)
∈
Ob(B(G)
0
),
we
have
×
O
×
(A)
=
O
(A)
=
k
L
where
k
L
denotes
the
algebraic
closure
of
k
in
L
[so
k
L
is
a
finite
separable
extension
of
k].
Theorem
6.2.
(Geometric
Frobenioids)
For
i
=
1,
2,
let
V
i
be
a
proper
normal
[geometrically
integral]
variety
over
a
field
k
i
;
K
i
the
function
field
of
V
i
;
i
/K
i
);
D
i
def
i
/K
i
a
[possibly
infinite]
Galois
extension;
G
i
def
=
Gal(
K
=
B(G
i
)
0
;
K
i
-Q-Cartier
set
of
prime
divisors
on
V
i
.
For
Spec(L
i
)
∈
Ob(D
i
),
D
K
i
=
∅
a
K
write
V
i
[L
i
]
for
the
normalization
of
V
i
in
L
i
;
D
L
i
for
the
set
of
prime
divisors
of
V
i
[L
i
]
that
map
into
[possibly
subvarieties
of
codimension
≥
1
of
]
prime
divisors
of
D
K
i
;
Φ
i
(L
i
)
⊆
Z
≥0
[D
L
i
]
⊆
Z[D
L
i
]
for
the
monoid
of
Cartier
effective
divisors
on
V
i
[L
i
]
with
support
in
D
L
i
;
B
i
(L
i
)
⊆
L
×
i
for
the
group
of
rational
functions
on
V
i
[L
i
]
whose
zeroes
and
poles
are
supported
on
D
L
i
;
B
i
(L
i
)
→
Φ
i
(L
i
)
gp
for
the
natural
map;
C
i
for
the
associated
model
Frobenioid
of
Theorem
5.2,
(ii).
Then:
(i)
Let
ψ
:
V
2
→
V
1
be
a
dominant
morphism
of
schemes
such
that
the
following
conditions
are
sat-
isfied:
(a)
D
K
2
is
equal
to
the
set
of
prime
divisors
of
V
2
that
map
into
a
prime
divisor
of
D
K
1
;
(b)
the
resulting
inclusion
of
function
fields
K
1
→
K
2
satisfies
the
2
factors
through
K
1
;
(c)
K
1
condition
that
the
composite
inclusion
K
1
→
K
2
→
K
is
separably
closed
in
K
2
.
Then
ψ
induces
a
functor
Ψ
:
C
1
→
C
2
THE
GEOMETRY
OF
FROBENIOIDS
I
111
[well-defined
up
to
isomorphism]
that
is
compatible
with
Frobenius
degrees,
the
functor
D
1
→
D
2
induced
by
the
inclusion
of
fields
K
1
→
K
2
,
and
the
natural
transformations
Φ
1
→
Φ
2
|
D
1
,
B
1
→
B
2
|
D
1
induced
by
pulling
back
divisors
and
rational
functions,
respectively,
via
ψ.
(ii)
Assume
that
the
data
labeled
by
the
index
“1”
is
equal
to
the
data
labeled
by
the
index
“2”
[so
in
the
following,
we
shall
omit
these
indices].
Also,
let
us
suppose
that
k
is
of
positive
characteristic
p.
Then
the
Frobenius
morphism
ψ
:
V
→
V
satisfies
the
conditions
of
(i),
hence
determines
a
functor
Ψ:
C→C
which
is
isomorphic
to
the
naive
Frobenius
functor
[of
degree
p
on
C]
of
Propo-
sition
2.1.
(iii)
We
maintain
the
assumption
of
(ii)
concerning
indices.
Then
the
Frobe-
nioid
C
is
of
isotropic,
standard,
and
birationally
Frobenius-normalized
of
type,
but
not
of
group-like
type.
If,
moreover,
for
every
finite
extension
L
⊆
K
K,
and
every
D
∈
D
L
,
it
holds
that
D
lies
in
the
support
of
the
image
in
Φ(L)
gp
of
an
element
of
B(L),
then
C
is
of
rationally
standard
type.
(iv)
We
maintain
the
assumption
of
(ii)
concerning
indices.
Then
D
is
Frobe-
nius-slim.
Let
Z
⊆
G
be
the
subgroup
of
elements
that
commute
with
some
open
subgroup
of
G.
Then
D
is
slim
if
and
only
if
Z
=
{1};
D
is
Div-slim
[relative
to
Φ]
if
and
only
if,
for
every
1
=
z
∈
Z,
there
exists
a
finite
Galois
extension
L
⊆
K
of
K
such
that
z
acts
nontrivially
on
Φ(L).
Proof.
First,
we
consider
assertion
(i).
Note
that
by
assumptions
(b),
(c)
[in
1
of
K
1
the
statement
of
assertion
(i)],
it
follows
that
any
finite
extension
L
1
⊆
K
def
2
of
K
2
such
that
[L
2
:
K
2
]
=
[L
1
:
determines
a
finite
extension
L
2
=
L
1
·
K
2
⊆
K
K
1
].
Thus,
ψ
determines
a
functor
D
1
→
D
2
.
Moreover,
by
assumption
(a)
[in
the
statement
of
assertion
(i)],
it
follows
that
by
pulling
back
[Cartier]
divisors
and
rational
functions
via
ψ,
we
obtain
compatible
natural
transformations
Φ
1
→
Φ
2
|
D
1
,
B
1
→
B
2
|
D
1
.
Thus,
it
follows
formally
from
the
definition
of
the
category
underlying
a
model
Frobenioid
in
Theorem
5.2,
(i),
that
we
obtain
a
functor
Ψ
:
C
1
→
C
2
satisfying
the
properties
stated
in
assertion
(i).
From
this
definition
of
the
functor
Ψ,
it
then
follows
immediately
from
the
definition
of
the
“Frobenius
morphism
in
characteristic
p”,
together
with
the
definition
of
the
“naive
Frobenius
functor”
of
Proposition
2.1
—
i.e.,
in
a
word,
that
both
functors
are
obtained
by
“raising
to
the
p-th
power”
—
that
these
two
functors
are
isomorphic.
This
completes
the
proof
of
assertions
(i),
(ii).
Next,
we
consider
assertion
(iii).
The
fact
that
C
is
of
isotropic
and
birationally
Frobenius-normalized
type
follows
formally
from
Theorem
5.2,
(ii).
The
fact
that
C
is
not
of
group-like
type
is
immediate
from
our
assumption
that
D
K
=
∅
[and
the
definition
of
Φ].
It
is
immediate
that
every
monomorphism
of
D
is
an
isomorphism,
hence
that
D
is
of
FSM-type
[hence
also
of
FSMFF-type
—
cf.
§0].
If
a
K-linear
of
K
induces
an
automorphism
of
Φ(L)
automorphism
α
of
a
finite
extension
L
⊆
K
112
SHINICHI
MOCHIZUKI
which
preserves
the
primes
of
L,
then
it
is
immediate
from
the
fact
that
α
induces
an
automorphism
of
the
scheme
V
[L]
that
α
maps
every
prime
divisor
D
∈
Φ(L)
to
D
[i.e.,
not
to
some
n
·
D,
where
n
≥
2];
thus,
we
conclude
that
Φ
is
non-dilating,
hence
that
C
is
of
standard
type.
Now
suppose
that
for
every
finite
extension
L
⊆
K
gp
of
K,
and
every
D
∈
D
L
,
it
holds
that
D
lies
in
the
support
of
the
image
in
Φ(L)
of
an
element
of
B(L).
Then
it
follows
formally
[cf.
Definition
4.5,
(ii)]
that
C
is
of
[strictly]
rational
type
[since
Φ
has
already
been
observed
to
be
perf-factorial
—
cf.
Example
6.1].
Thus,
C
satisfies
condition
(a)
of
Definition
4.5,
(iii).
Now
I
claim
that
every
object
of
(C
un-tr
)
birat
is
Frobenius-compact.
Indeed,
if
α
is
a
K-linear
of
K
that
acts
by
multiplication
by
automorphism
of
a
finite
extension
L
⊆
K
birat
pf
λ
∈
Q
>0
on
Φ
(L)
(
=
0),
then
since
α
induces
an
automorphism
of
the
variety
V
[L],
it
follows
that
the
order
∈
Q
>0
of
the
zero
[or
pole]
of
highest
order
of
an
element
f
∈
Φ
birat
(L)
pf
is
preserved
by
α,
hence
that
λ
=
1.
This
completes
the
proof
of
the
claim,
and
hence
of
the
fact
that
C
is
of
rationally
standard
type.
is
a
finite
Finally,
we
consider
assertion
(iv).
First,
we
observe
that
if
L
⊆
K
extension
of
K
that
corresponds
to
an
open
subgroup
H
⊆
G,
then
there
is
a
natural
isomorphism
∼
(Z
⊇)
Z
G
(H)
→
Aut(D
Spec(L)
→
D)
[cf.
[Mzk7],
Corollary
1.1.6].
Since
G
is
profinite,
hence,
in
particular,
residually
finite,
it
follows
formally
that
Z,
Z
G
(H)
are
also
residually
finite,
hence
that
D
is
Frobenius-slim,
by
Remark
3.1.2.
Also,
since
Z
is
the
union
of
subgroups
of
G
of
the
form
“Z
G
(H)”,
it
follows
formally
that
D
is
slim
if
and
only
if
Z
=
{1},
and
that
D
is
Div-slim
[relative
to
Φ]
if
and
only
if,
for
every
1
=
z
∈
Z,
there
exists
of
K
such
that
z
acts
nontrivially
on
Φ(L
).
This
a
finite
Galois
extension
L
⊆
K
completes
the
proof
of
assertion
(iv).
Remark
6.2.1.
Theorem
6.2,
(ii),
constitutes
the
principal
justification
for
the
name
“Frobenius
functor”
that
was
applied
to
various
functors
in
§2.
From
this
point
of
view,
the
decomposition
of
the
naive
Frobenius
functor
of
Proposition
2.1
into
“unit-linear”
and
“unit-wise”
Frobenius
functors
[cf.
the
proof
of
Corollary
2.6]
may
be
thought
of
as
corresponding
to
the
decomposition
of
the
Frobenius
morphism
in
positive
characteristic
algebraic
geometry
over
a
fixed
base
into
the
composite
of
a
“relative
Frobenius
morphism”,
which
is
linear
over
the
fixed
base,
with
the
Frobenius
morphism
of
the
fixed
base.
Example
6.3.
A
Frobenioid
of
Arithmetic
Origin.
Let
F
be
a
number
field
[cf.
§0].
Write
V(F
)
for
the
set
of
valuations
on
F
[where
we
identify
com-
plex
archimedean
valuations
with
their
complex
conjugates];
O
F
for
the
ring
of
integers
of
F
.
If
v
∈
V(F
),
then
we
shall
write
F
v
for
the
completion
of
F
at
v;
O
v
×
⊆
F
v
×
for
the
group
of
units
[i.e.,
elements
of
valuation
1
of
F
v
×
];
O
v
⊆
F
v
×
for
the
multiplicative
monoid
of
elements
of
valuation
≤
1;
μ(F
)
⊆
O
F
×
for
the
def
def
group
of
roots
of
unity
in
F
;
ord(F
v
)
=
F
v
×
/O
v
×
;
ord(O
v
)
=
O
v
/O
v
×
⊆
ord(F
v
).
Thus,
ord(F
v
)
=
ord(O
v
)
gp
;
ord(F
v
)
∼
=
Z,
ord(O
v
)
∼
=
Z
≥0
if
v
is
nonarchimedean;
THE
GEOMETRY
OF
FROBENIOIDS
I
113
ord(F
v
)
∼
=
R,
ord(O
v
)
∼
=
R
≥0
if
v
is
archimedean.
We
shall
refer
to
an
element
of
the
monoid
def
ord(O
v
)
Φ(F
)
=
v∈V(F
)
as
an
effective
arithmetic
divisor
on
F
,
and
to
an
element
of
the
group
Φ(F
)
gp
=
ord(F
v
)
v∈V(F
)
as
an
arithmetic
divisor
on
F
.
Thus,
there
is
a
natural
homomorphism
of
groups
def
B(F
)
=
F
×
→
Φ(F
)
gp
[given
by
mapping
an
element
f
∈
F
×
to
the
images
of
f
in
the
various
factors
F
v
×
/O
v
×
=
ord(F
v
),
all
but
a
finite
number
of
which
are
zero].
Note,
moreover,
that
Φ,
B,
as
well
as
the
homomorphism
B
→
Φ
gp
are
functorial
in
the
number
field
def
F
.
Thus,
if
F
is
a
[not
necessarily
finite]
Galois
extension
of
F
,
G
=
Gal(
F
/F
),
def
D
=
B(G)
0
,
then
Φ,
B
determine
monoids
on
D,
and
we
have
a
natural
homo-
morphism
B
→
Φ
gp
.
Moreover,
for
each
finite
extension
L
⊆
F
of
F
,
one
veri-
fies
immediately
that
Φ(L)
=
0
is
perf-factorial,
that
there
is
a
natural
bijection
∼
Prime(Φ(L))
→
V(L),
and
that
the
supports
of
elements
of
Φ(L)
are
precisely
the
finite
subsets
of
V(L).
Thus,
by
Theorem
5.2,
(ii),
this
data
determines
a
[model]
Frobenioid
C
F
/F
of
isotropic
and
birationally
Frobenius-normalized
type.
Note
that
an
object
of
C
F
/F
that
projects
to
Spec(L)
∈
Ob(B(G)
0
)
may
be
thought
of
as
an
arithmetic
line
bundle
L
on
L
[i.e.,
a
line
bundle
on
Spec(O
L
),
equipped
with
Hermitian
metrics
at
the
archimedean
primes
—
cf.
[Szp],
pp.
13-14].
If
L
is
an
arithmetic
line
bundle
on
L,
and
M
is
an
arithmetic
line
bundle
on
M
[where
M
⊆
F
is
a
finite
extension
of
F
],
then
one
verifies
immediately
that
a
morphism
L
→
M
in
C
F
/F
may
be
thought
of
as
consisting
of
the
following
data:
(a)
a
morphism
Spec(L)
→
Spec(M
)
over
Spec(F
);
(b)
an
element
d
∈
N
≥1
;
(c)
a
nonzero
morphism
of
arithmetic
line
bundles
L
⊗d
→
M|
L
on
L.
Also,
we
observe
that
for
A
∈
Ob(C
F
/F
)
that
projects
to
Spec(L)
∈
Ob(B(G)
0
),
we
have
O
×
(A)
=
O
(A)
=
μ(L)
[cf.,
for
instance,
[Szp],
p.
15].
Also,
observe
that
we
have
a
natural
arithmetic
degree
homomorphism
:
Φ(L)
gp
→
R
deg
arith
L
obtained
as
follows:
If
v
is
archimedean,
so
we
have
a
natural
embedding
of
topo-
logical
fields
R
→
F
v
,
then
the
restriction
of
deg
arith
to
the
factor
ord(F
v
)
maps
L
the
image
of
λ
∈
R
>0
to
−[F
v
:
R]
·
log(λ).
If
v
is
nonarchimedean,
then
the
re-
striction
of
deg
arith
to
the
factor
ord(F
v
)
maps
the
image
of
an
element
λ
∈
O
v
to
L
114
SHINICHI
MOCHIZUKI
the
natural
logarithm
of
the
cardinality
of
the
finite
set
O
v
/(λ)
[where
O
v
is
the
vanishes
on
the
ring
of
integers
of
F
v
].
Thus,
one
verifies
immediately
that
deg
arith
L
image
of
B(L).
Remark
6.3.1.
In
light
of
Examples
6.1,
6.3,
many
readers
might
expect
that
the
next
natural
step
is
to
attempt
to
apply
the
theory
of
Frobenioids
to
study
arith-
metic
line
bundles
on
higher-dimensional
arithmetic
varieties.
This
leads,
however,
to
numerous
complications
which
are
beyond
the
scope
of
the
present
paper.
More-
over,
it
is
not
even
clear
to
the
author
at
the
time
of
writing
that
this
constitutes
a
natural
direction
in
which
to
further
develop
the
theory
of
Frobenioids.
Theorem
6.4.
(Arithmetic
Frobenioids)
For
i
=
1,
2,
let
F
i
be
a
number
def
def
field;
F
i
/F
i
a
[possibly
infinite]
Galois
extension;
G
i
=
Gal(
F
i
/F
i
);
D
i
=
B(G
i
)
0
;
Φ
i
the
monoid
on
D
i
given
by
the
effective
arithmetic
divisors;
B
i
the
group-like
monoid
on
D
i
given
by
the
multiplicative
group
of
the
field
extension
of
F
i
in
question;
B
i
→
Φ
gp
i
the
natural
map;
C
i
the
associated
model
Frobenioid
of
Theorem
5.2,
(ii).
Then:
(i)
Assume
that
the
data
labeled
by
the
index
“1”
is
equal
to
the
data
labeled
by
the
index
“2”
[so
in
the
remainder
of
the
present
assertion
(i),
we
shall
omit
these
indices].
Then
the
Frobenioids
C,
C
pf
,
C
rlf
,
C
un-tr
,
(C
pf
)
un-tr
are
of
isotropic
and
rationally
standard
type,
but
not
of
group-like
type;
D
is
Frobenius-slim
and
Div-slim
[with
respect
to
Φ,
Φ
pf
,
Φ
rlf
].
Moreover,
D
is
slim
if
and
only
if
the
subgroup
of
elements
of
G
that
commute
with
some
open
subgroup
of
G
is
trivial.
Finally,
if
A
∈
Ob(C
rlf
)
is
a
Frobenius-trivial
object
that
projects
to
the
object
of
D
determined
by
a
finite
extension
L
⊆
F
of
F
,
then
deg
arith
determines
an
L
isomorphism
of
groups
∼
δ
A
:
Pic
Φ
(A)
→
R
[cf.
Theorem
5.1,
(i)].
(ii)
Let
∼
Ψ
rlf
:
C
1
rlf
→
C
2
rlf
be
an
equivalence
of
categories
between
the
realifications
[cf.
Proposition
5.3]
of
C
1
,
C
2
.
Then
there
exists
an
element
deg(Ψ
rlf
)
∈
R
>0
such
that
for
all
Frobenius-trivial
A
1
∈
Ob(C
1
),
A
2
∈
Ob(C
2
)
such
that
A
2
=
Ψ
rlf
(A
1
)
[where
we
recall
that
Ψ
rlf
preserves
Frobenius-trivial
objects
—
cf.
(i);
Corollary
4.11,
(iv)],
∼
the
composite
of
δ
A
2
with
the
isomorphism
Pic
Φ
(A
1
)
→
Pic
Φ
(A
2
)
determined
by
Ψ
rlf
[cf.
(i)
above;
Corollary
4.10;
Corollary
4.11,
(iii)]
is
equal
to
deg(Ψ
rlf
)
·
δ
A
1
.
(iii)
If
the
equivalence
of
categories
Ψ
rlf
of
(ii)
arises
from
an
equivalence
of
categories
∼
(Ψ
pf
)
un-tr
:
(C
1
pf
)
un-tr
→
(C
2
pf
)
un-tr
THE
GEOMETRY
OF
FROBENIOIDS
I
115
between
the
unit-trivialized
perfections
of
C
1
,
C
2
[cf.
(i);
Corollary
5.4],
then
deg(Ψ
rlf
)
∈
Q
>0
.
In
particular,
if
A
1
∈
Ob((C
1
pf
)
un-tr
)
[whose
projection
to
D
1
we
denote
by
Spec(L
1
)],
A
2
∈
Ob((C
2
pf
)
un-tr
)
[whose
projection
to
D
2
we
denote
by
Spec(L
2
)],
A
2
=
(Ψ
pf
)
un-tr
(A
1
),
then
the
bijection
∼
∼
∼
V(L
1
)
→
Prime(Φ
1
(L
1
))
→
Prime(Φ
2
(L
2
))
→
V(L
2
)
induced
by
(Ψ
pf
)
un-tr
[cf.
(i);
Corollary
4.11,
(iii)]
maps
a
valuation
v
1
∈
V(L
1
)
lying
over
a
valuation
v
0
of
Q
to
a
valuation
v
2
∈
V(L
2
)
lying
over
the
valuation
v
0
of
Q.
(iv)
If
the
equivalence
of
categories
Ψ
rlf
of
(ii)
arises
from
an
equivalence
of
categories
∼
Ψ
:
C
1
→
C
2
between
C
1
,
C
2
[cf.
(i);
(iii);
Theorem
3.4,
(iii),
(iv)],
then
deg(Ψ
rlf
)
=
1.
If,
moreover,
there
exists
a
finite
extension
L
1
⊆
F
1
of
F
1
which
is
Galois
over
Q,
∼
then
the
corresponding
[i.e.,
via
the
equivalence
D
1
→
D
2
induced
by
Ψ
—
cf.
(i);
Corollary
4.11,
(ii)]
finite
extension
L
2
⊆
F
2
of
F
2
is
isomorphic
to
L
1
in
a
fashion
that
is
compatible
with
an
isomorphism
F
1
∼
=
F
2
.
Proof.
First,
we
consider
assertion
(i).
We
have
already
seen
in
Example
6.3
that
the
Frobenioid
C
is
of
isotropic
and
birationally
Frobenius-normalized
type,
and
that
Φ
is
nonzero
[so
C
is
not
of
group-like
type]
and
perf-factorial.
As
was
observed
in
the
proof
of
Theorem
6.2,
(iii),
(iv),
D
is
Frobenius-slim
and
of
FSM-type,
hence
also
of
FSMFF-type.
Moreover,
since
any
automorphism
of
a
number
field
that
fixes
all
of
the
valuations
of
the
number
field
is
clearly
equal
to
the
identity
automorphism,
it
follows
immediately
that
Φ
is
non-dilating,
and
that
D
is
Div-slim
[relative
to
Φ,
hence
also
relative
to
Φ
pf
,
Φ
rlf
].
Also,
it
is
immediate
from
the
definition
of
B
that
C
is
of
[strictly]
rational
type,
and
that
every
object
of
(C
un-tr
)
birat
is
Frobenius-
compact.
Thus,
we
conclude
that
C
[hence
also
C
pf
,
C
rlf
,
C
un-tr
,
(C
pf
)
un-tr
—
cf.
Proposition
5.5,
(iii)]
is
of
rationally
standard
type.
The
proof
of
the
criterion
for
D
to
be
slim
is
entirely
similar
to
the
proof
given
for
Theorem
6.2,
(iv).
Finally,
to
show
that
the
surjection
δ
A
:
Pic
Φ
(A)
R
is,
in
fact,
an
isomorphism,
it
suffices
to
verify
that
the
image
of
Φ
birat
(L)
⊗
Z
R
=
gp
rlf
gp
(L
×
)⊗
Z
R
in
(Φ
rlf
factor
)
(L)
is
equal
to
the
set
of
elements
of
(Φ
factor
)
(L)
with
finite
support
whose
image
under
deg
arith
is
0.
But
this
is
an
immediate
consequence
of
L
the
well-known
Dirichlet
unit
theorem
of
classical
number
theory
[cf.,
e.g.,
[Lang2],
p.
104].
This
completes
the
proof
of
assertion
(i).
Now
assertion
(ii)
follows
by
observing
that
the
isomorphism
of
groups
∼
Pic
Φ
(A
1
)
→
Pic
Φ
(A
2
)
determined
by
Ψ
rlf
[cf.
assertion
(i);
Corollary
4.10;
Corollary
4.11,
(iii)]
is
compat-
ible
with
the
“order
structure”
induced
on
both
sides
[via
δ
A
1
,
δ
A
2
]
by
the
“order
116
SHINICHI
MOCHIZUKI
structure”
of
R.
[Indeed,
this
compatibility
follows
from
the
fact
that
the
isomor-
∼
rlf
phism
in
question
arises
from
an
isomorphism
of
monoids
Φ
rlf
1
(A
1
)
→
Φ
2
(A
2
).]
This
completes
the
proof
of
assertion
(ii).
Next,
we
observe
that
assertion
(iii)
follows
formally
from
assertion
(ii),
by
applying
Lemma
6.5,
(ii),
below
in
the
following
fashion:
If
deg(Ψ
rlf
)
∈
Q
>0
,
then
one
verifies
immediately
that
there
exist
three
nonarchimedean
valuations
w
1
,
w
3
,
w
5
∈
V(L
1
)
lying
over
primes
p
1
,
p
3
,
p
5
∈
Primes,
respectively,
with
the
property
that
w
1
→
w
2
∈
V(L
2
),
w
3
→
w
4
∈
V(L
2
),
w
5
→
w
6
∈
V(L
2
),
where
w
2
,
w
4
,
w
6
lie
over
primes
p
2
,
p
4
,
p
6
∈
Primes,
respectively,
such
that
p
1
,
p
2
,
p
3
,
p
4
,
p
5
,
p
6
are
distinct.
But
this
implies
that
log(p
1
)/log(p
2
),
log(p
3
)/log(p
4
),
log(p
5
)/log(p
6
)
∈
(deg(Ψ
rlf
))
−1
·
Q
>0
in
contradiction
to
Lemma
6.5,
(ii).
Thus,
deg(Ψ
rlf
)
∈
Q
>0
.
The
final
portion
of
assertion
(iii)
concerning
valuations
of
Q
now
follows
from
Lemma
6.5,
(i).
This
completes
the
proof
of
assertion
(iii).
Finally,
we
consider
assertion
(iv).
Suppose
that
v
1
∈
V(L
1
)
maps
to
v
2
∈
V(L
2
)
[cf.
the
notation
of
the
statement
of
assertion
(iii)].
For
i
=
1,
2,
write
deg(L
i
,
v
i
)
for
the
number
of
valuations
∈
V(L
i
),
including
v
i
,
that
lie
over
the
same
valuation
of
Q
as
v
i
.
Then
by
Tchebotarev’s
density
theorem
[cf.,
e.g.,
[Lang2],
Chapter
VIII,
§4,
Theorem
10],
it
follows
that
[L
i
:
Q]
is
equal
to
the
maximum
of
the
deg(L
i
,
v
i
),
as
v
i
ranges
over
the
elements
of
V(L
i
).
Moreover,
if
v
i
is
nonarchimedean
and
lies
over
a
prime
p
i
∈
V(L
i
),
then
p
i
splits
completely
in
L
i
if
and
only
if
deg(L
i
,
v
i
)
=
[L
i
:
Q].
Thus,
it
follows
that
if
v
1
,
v
2
lie
over
a
prime
p
∈
Primes
[cf.
assertion
(iii)],
then
[again
by
assertion
(iii)]
p
splits
completely
in
L
1
if
and
only
if
p
splits
completely
in
L
2
.
If
this
is
the
case,
then
it
follows
that
deg
arith
maps
a
generator
L
i
of
the
monoid
Φ
i
(L
i
)
v
i
(
∼
Z
)
to
log(p).
Thus,
we
conclude
that
deg(Φ
rlf
)
=
1,
as
=
≥0
desired.
Note
that
this
implies
that
v
1
is
of
degree
1
[i.e.,
deg
arith
maps
a
generator
L
1
∼
of
the
monoid
Φ
1
(L
1
)
v
1
(
=
Z
≥0
)
to
log(p)]
if
and
only
if
v
2
is
of
degree
1.
Thus,
if
L
1
is
Galois
over
Q,
then
whenever
v
2
is
of
degree
1,
it
follows
that
v
1
is
of
degree
1,
hence
that
p
splits
completely
in
L
1
[since
L
1
is
Galois
over
Q].
But
this
implies
[again
by
Tchebotarev’s
density
theorem
—
cf.,
e.g.,
[NSW],
Theorem
12.2.5]
that
L
1
⊆
L
2
,
hence
that
L
1
=
L
2
[since
we
have
already
seen
that
[L
1
:
Q]
=
[L
2
:
Q]].
This
completes
the
proof
of
assertion
(iv).
Lemma
6.5.
Numbers)
(Transcendental
Properties
of
Logarithms
of
Prime
(i)
The
real
numbers
log(p)
∈
R,
where
p
ranges
over
the
prime
numbers,
are
linearly
independent
over
Q.
(ii)
Let
p
1
,
p
2
,
.
.
.
,
p
6
be
distinct
prime
numbers.
Then
there
do
not
exist
λ
1
,
λ
2
∈
Q
>0
such
that:
log(p
1
)/log(p
2
)
=
λ
1
·log(p
3
)/log(p
4
)
=
λ
2
·log(p
5
)/log(p
6
).
THE
GEOMETRY
OF
FROBENIOIDS
I
117
Proof.
Assertion
(i)
is
a
formal
consequence
of
the
fact
that
Z
is
a
unique
factor-
ization
domain.
Assertion
(ii)
is
a
consequence
of
a
theorem
of
Lang
[cf.
[Lang1];
[Baker],
p.
119]:
Indeed,
since
the
log(p
i
)
are
linearly
independent
over
Q
[by
as-
sertion
(i)],
it
follows
that
each
of
the
following
two
sets
of
numbers
is
also
linearly
independent
over
Q:
{log(p
2
),
log(p
4
),
log(p
6
)};
{1,
log(p
3
)/log(p
4
)}
Moreover,
all
six
products
of
one
element
from
the
first
set
and
one
element
from
the
second
set
are
of
the
form
μ
·
log(p
i
),
where
μ
∈
Q
>0
.
Thus,
the
exponential
of
each
of
these
products
is
algebraic,
in
contradiction
to
Lang’s
theorem.
118
SHINICHI
MOCHIZUKI
Appendix:
Slim
Exponentiation
In
the
present
Appendix,
we
discuss
some
elementary
general
nonsense
con-
cerning
slim
categories.
Definition
A.1.
(i)
A
2-category
of
1-categories
will
be
called
2-slim
[cf.
[Mzk7],
Definition
1.2.4,
(iii)]
if
every
1-morphism
[i.e.,
functor]
in
the
2-category
has
no
nontrivial
automorphisms.
(ii)
If
D
is
a
2-category
of
1-categories,
then
we
shall
write
|D|
for
the
associated
1-category
whose
objects
are
objects
of
D
and
whose
morphisms
are
isomorphism
classes
of
morphisms
of
D
[cf.
[Mzk7],
Definition
1.2.4,
(iv)].
We
shall
also
refer
to
|D|
as
the
coarsification
of
C.
Remark
A.1.1.
The
name
“coarsification”
is
motivated
by
the
theory
of
“coarse
moduli
spaces”
associated
to
(say)
“fine
moduli
stacks”
[cf.
[Mzk7],
Remark
1.2.4.1].
The
following
result
may
be
regarded
as
a
generalization
of
[Mzk7],
Proposition
1.2.5,
(ii)
[a
result
concerning
anabelioids],
to
the
case
of
arbitrary
slim
categories.
Proposition
A.2.
(Slim
Exponentiation)
Let
C
be
a
slim
category
[cf.
§0].
Let
D
be
the
2-category
of
1-categories
defined
as
follows:
The
objects
of
D
are
the
categories
C
A
[cf.
§0],
where
A
∈
Ob(A).
The
1-morphisms
of
D
are
the
functors
f
!
:
C
A
→
C
B
[cf.
§0]
induced
by
morphisms
f
:
A
→
B
of
C.
The
2-morphisms
of
D
are
isomorphisms
between
these
functors
[cf.
§0].
Then
D
is
2-slim.
Moreover,
the
functor
E
:
C
→
|D|
A
→
C
A
;
f
→
f
!
∼
determines
an
equivalence
of
categories
C
→
|D|.
We
shall
refer
to
the
functor
E
as
the
slim
exponentiation
functor.
Proof.
The
fact
that
D
is
2-slim
follows
immediately
from
the
assumption
that
C
is
slim.
Now
it
is
immediate
from
the
definitions
that
E
is
full
and
essentially
surjective.
To
verify
that
E
is
faithful,
let
us
first
observe
that
given
any
two
morphisms
f,
g
:
A
→
B
of
C,
an
isomorphism
between
the
functors
f
!
,
g
!
:
C
A
→
C
B
determines
an
isomorphism
between
the
composites
of
the
functors
f
!
,
g
!
with
THE
GEOMETRY
OF
FROBENIOIDS
I
119
the
natural
functor
C
B
→
C.
On
the
other
hand,
these
two
composite
functors
C
A
→
C
both
coincide
with
the
natural
functor
C
A
→
C
[i.e.,
that
maps
an
object
∼
C
→
A
of
C
A
to
the
object
C
of
C].
Thus,
any
isomorphism
f
!
→
g
!
determines
an
automorphism
of
the
natural
functor
C
A
→
C,
which
[by
the
slimness
of
C!]
is
the
∼
identity
automorphism.
But
this
implies
[by
applying
the
isomorphism
f
!
→
g
!
to
id
A
the
object
A
−→A
of
C
A
]
that
f
=
g,
as
desired.
120
SHINICHI
MOCHIZUKI
Index
1-commutative,
§0
2-slim,
A.1,
(i)
abstract
equivalence,
§0
almost
totally
epimorphic,
§0
anchor,
§0
A-pair,
proof
of
5.1,
(i)
arithmetic
degree,
6.3
arithmetic
divisor,
6.3
arithmetic
line
bundle,
6.3
Aut-ample,
1.2,
(iv),
(v)
Aut-saturated,
§0
Aut
sub
-ample,
1.2,
(iv),
(v)
Aut
sub
-saturated,
§0
Aut-type,
§0
base
category,
1.1,
(iii),
(iv)
base-equivalent,
1.2,
(ii)
base-Frobenius
pair
(of
a
Frobenioid),
2.7,
(iii)
base-Frobenius
pair
(of
a
Frobenius-trivial
object),
5.6
base-FSM-morphism,
1.2,
(ii)
base-identity,
1.2,
(ii)
base-isomorphism
(base-isomorphic),
1.2,
(ii)
base-section,
2.7,
(i)
base-trivial,
1.2,
(iv),
(v)
birationalization
(of
a
Frobenioid),
4.4
birationally
Frobenius-normalized,
4.5,
(i)
bounded,
§0
categorical
fiber
product,
§0
categorical
quotient,
§0
centralizer,
§0
characteristic,
§0
characteristically
injective,
§0
characteristic
splitting,
2.3
characteristic
type,
§0
co-angular,
1.2,
(iii)
coarsification,
A.1,
(ii)
connected
category,
§0
co-objective,
§0
co-primary,
4.1,
(iii)
co-prime
type,
2.8,
(iii)
Div-equivalent,
1.2,
(ii)
Div-Frobenius-trivial,
1.2,
(iv),
(v)
Div-identity,
1.2,
(ii)
THE
GEOMETRY
OF
FROBENIOIDS
I
divisorial,
1.1,
(i)
divisor
monoid,
1.1,
(iv);
5.2,
(ii)
Div-slim,
4.5,
(iv)
effective
arithmetic
divisor,
6.3
elementary
Frobenioid,
1.1,
(iii)
End-ample,
1.2,
(iv),
(v)
End-equivalence,
§0
factorization
homomorphism,
2.4,
(i),
(c)
factorization
of
morphisms
of
a
Frobenioid,
1.3,
(iv),
(a)
factorization
of
pre-steps
of
a
Frobenioid,
1.3,
(v),
(b),
(c)
F
-distinguished,
2.7,
(ii)
fiberwise
surjective,
§0
finitely
(respectively,
countably)
connected
type,
§0
F
P-path,
proof
of
5.2,
(iv)
Frobenioid,
1.3
Frobenius-ample,
1.2,
(iv),
(v)
Frobenius-compact,
1.2,
(iv),
(v)
Frobenius
degree,
1.1,
(iii),
(iv)
Frobenius
functor
(on
an
elementary
Frobenioid),
2.4,
(iii)
Frobenius-isotropic,
1.2,
(iv),
(v)
Frobenius-normalized,
1.2,
(iv),
(v)
Frobenius-section,
2.7,
(ii)
Frobenius-slim,
3.1,
(i)
Frobenius-trivial,
1.2,
(iv),
(v)
FSMFF-type
(category
of),
§0
FSMI-morphism,
§0
FSM-morphism,
§0
FSM-type
(category
of),
§0
groupification,
§0
group-like
(monoid),
1.1,
(i)
group-like
(object
of
a
pre-Frobenioid),
1.2,
(iv),
(v)
immobile,
§0
integral,
§0
irreducible
(element
of
a
monoid),
§0
irreducible
(morphism
of
a
category),
§0
isometric
morphism
(isometry),
1.2,
(i)
iso-subanchor,
§0
isotropic,
1.2,
(iv),
(v)
isotropic
hull,
1.2,
(iv)
isotropification
functor,
1.9,
(v)
K-Q-Cartier,
6.1
LB-invertible,
1.2,
(iii)
left-hand
isomorphism,
4.2,
(iii)
linear
morphism,
1.2,
(i)
121
122
SHINICHI
MOCHIZUKI
metrically
equivalent,
1.2,
(i)
metrically
trivial,
1.2,
(iv),
(v)
mid-adjoint,
§0
minimal-adjoint,
§0
minimal-coadjoint,
§0
mobile,
§0
model
Frobenioid,
5.2,
(ii)
model
type,
4.5,
(i)
monoid,
§0
monoid
(on
a
category),
1.1,
(ii)
monoid
type,
§0
mono-minimal,
§0
monoprime,
§0
morphism
of
Frobenius
type,
1.2,
(iii)
naive
Frobenius
functor,
2.1,
(i)
natural
projection
functor,
1.1,
(iii),
(iv)
non-dilating
(endomorphism),
1.1,
(i)
non-dilating
(monoid
on
a
category),
1.1,
(ii)
number
field,
§0
one-morphism
category,
§0
one-object
category,
§0
opposite
category,
§0
(p
1
,
p
2
)-admissible,
proof
of
3.4
P-distinguished,
2.7,
(i)
perfection
(of
a
Frobenioid),
3.1,
(iii)
perfection
(of
a
monoid),
§0
perfect
(monoid),
§0
perfect
(object
of
a
pre-Frobenioid),
1.2,
(iv)
perf-factorial,
2.4,
(i)
pre-divisorial,
1.1,
(i)
pre-Frobenioid,
1.1,
(iv)
pre-Frobenioid
structure,
1.1,
(iv)
pre-model
type,
2.7,
(iii)
pre-step,
1.2,
(iii)
primary
(element
of
a
monoid),
§0
primary
(pre-step),
1.2,
(iii)
prime,
§0
prime-Frobenius
morphism,
1.2,
(iii)
pro-l
portion,
2.8,
(ii)
pseudo-terminal,
§0
pull-back
morphism,
1.2,
(ii)
quasi-base-Frobenius
pair
(of
a
Frobenioid),
2.7,
(iii)
quasi-base-Frobenius
pair
(of
a
Frobenius-trivial
object),
5.6
quasi-connected,
§0
quasi-Frobenius-section,
2.7,
(ii)
THE
GEOMETRY
OF
FROBENIOIDS
I
quasi-Frobenius-trivial,
1.2,
(iv),
(v)
quasi-isotropic
type,
3.1,
(i)
raising
to
the
ζ-th
power,
2.8,
(iii)
rational
function
monoid,
4.4,
(ii);
5.2,
(ii)
rationally
standard
type,
4.5,
(iii)
rational
object,
4.5,
(ii)
rational
type,
4.5,
(ii)
realification
(of
a
Frobenioid),
5.3
realification
(of
a
perf-factorial
monoid),
2.4,
(i)
residually
finite
group,
3.1.2
right-hand
isomorphism,
4.2,
(iii)
rigid,
§0
saturated,
§0
sharp,
§0
skeletal
subcategory,
§0
skeleton,
§0
slim
(category),
§0
slim
exponentiation
functor,
A.2
slim
(profinite
group),
§0
standard
Frobenioid,
1.1,
(iii)
standard
type,
3.1,
(i)
step,
1.2,
(iii)
strictly
rational
object,
4.5,
(ii)
strictly
rational
type,
4.5,
(ii)
subanchor,
§0
sub-automorphism,
§0
subordinate,
§0
sub-quasi-Frobenius-trivial,
1.2,
(iv),
(v)
supporting
monoid
type,
2.4,
(ii)
support
(of
an
element
of
a
perf-factorial
monoid),
2.4,
(i),
(d)
supremum,
§0
terminal,
§0
totally
epimorphic,
§0
twin-primary,
proof
of
4.9
unit-equivalence,
3.1,
(iv)
unit-linear
Frobenius
functor,
2.5,
(iii)
unit-profinite
type,
2.8,
(i)
unit-trivial,
1.2,
(iv),
(v)
unit-trivialization
(of
a
Frobenioid),
3.1,
(iv)
unit-wise
Frobenius
functor,
2.6,
2.9
universally
Div-Frobenius-trivial,
1.2,
(iv),
(v)
zero
divisor,
1.1,
(iii),
(iv)
123
124
SHINICHI
MOCHIZUKI
Chart
of
Types
of
Morphisms
in
a
Frobenioid
type
of
morphism
projection
to
base
zero
divisor
Frobenius
degree
linear
isometry
base-isomorphism
base-FSM-morphism
pull-back
morphism
pre-step
step
primary
pre-step
isometric
pre-step
LB-invertible
morphism
of
Frobenius
type
prime-
Frobenius
morphism
?
?
isomorphism
FSM-morphism
?
isomorphism
isomorphism
isomorphism
isomorphism
?
?
0
?
?
0
?
=0
primary
0
0
1
?
?
?
1
1
1
1
1
?
isomorphism
0
?
isomorphism
0
prime
THE
GEOMETRY
OF
FROBENIOIDS
I
125
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